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Moment of inertia
Classical mechanics |
Core topics |
The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation. It is an extensive (additive) property: the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). One of its definitions is the second moment of mass with respect to distance from an axis r, , integrating over the entire mass .
For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about an axis perpendicular to the plane. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix; each body has a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
When a body is rotating, or free to rotate, around an axis, a torque must be applied to change its angular momentum. The amount of torque needed for any given rate of change in angular momentum is proportional to the moment of inertia of the body. Moment of inertia may be expressed in terms of kilogram-square metres (kg·m2) in SI units and pound-square feet (lbm·ft2) in imperial or US units.
Moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by d2m, where d is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum.[1] The term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765,[1][2] and it is incorporated into Euler's second law.
The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.[3][4]
Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.[5][6]
The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axis determines how steering forces on the control surfaces of its wings, elevators and tail affect the plane in roll, pitch and yaw.
Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, that is[7][8]
If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster.[7][8][9][10][11][12][13]
If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque τ on a body to the angular acceleration α around a principal axis, that is
For a simple pendulum, this definition yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as,
Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation.
This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses dm each multiplied by the square of its perpendicular distance r to an axis S .
In general, given an object of mass m, an effective radius k can be defined for an axis through its center of mass, with such a value that its moment of inertia is
where k is known as the radius of gyration.
Simple pendulum
Moment of inertia can be measured using a simple pendulum, because it is the resistance to the rotation caused by gravity. Mathematically, the moment of inertia of the pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle m with the square of its distance r to the pivot, that is
This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torque around the axis perpendicular to the plane of the pendulum movement. Here r is the distance vector perpendicular to and from the force to the torque axis. Here F is the tangential component of the net force on the mass. Associated with this torque is an angular acceleration, , of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is . Since the torque equation becomes:
where e is a unit vector perpendicular to the plane of the pendulum. (The second to the last step occurs because of the BAC-CAB rule using the fact that is always perpendicular to r.) The quantity I = mr2 is the moment of inertia of this single mass around the pivot point.
The quantity I = mr2 also appears in the angular momentum of a simple pendulum, which is calculated from the velocity v = ω×r of the pendulum mass around the pivot, where ω is the angular velocity of the mass about the pivot point. This angular momentum is given by
using math similar to that used to derive the previous equation.
Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
This shows that the quantity I = mr2 is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values mr2 for all of the elements of mass in the body.
Compound pendulum
A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of.[14][15]:395–396[16]:51–53 The natural frequency () of a compound pendulum depends on its moment of inertia, ,
where is the mass of the object, is local acceleration of gravity, and is the distance from the pivot point to the centre of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.[17]:516–517
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (), to obtain
where is the period (duration) of oscillation (usually averaged over multiple periods).
The moment of inertia of the body about its centre of mass, , is then calculated using the parallel axis theorem to be
where is the mass of the body and is the distance from the pivot point to the centre of mass .
Moment of inertia of a body is often defined in terms of its radius of gyration, which is the radius of a ring of equal mass around the centre of mass of a body that has the same moment of inertia. The radius of gyration is calculated from the body's moment of inertia and mass as the length,[18]:1296–1297
Center of oscillation
A simple pendulum that has the same natural frequency as a compound pendulum defines the length from the pivot to a point called the centre of oscillation of the compound pendulum. This point also corresponds to the centre of percussion. The length is determined from the formula,
The seconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of π radians/second for the pendulum. In this case, the distance to the center of oscillation, , can be computed to be
Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.
Measuring moment of inertia
The moment of inertia of complex systems such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.[19] The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.[20]
Calculating moment of inertia about an axis
The moment of inertia about an axis of a body is calculated by summing mr2 for every particle in the body, where r is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.[21])
Consider the kinetic energy of an assembly of N masses mi that lie at the distances ri from the pivot point P, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,[17]:516–517[18]:1084–1085 [18]:1296–1300
This shows that the moment of inertia of the body is the sum of each of the mr2 terms, that is
Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:
Another expression replaces the summation with an integral,
Here, the function ρ gives the mass density at each point (x, y, z), r is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point (x, y, z) in the solid, and the integration is evaluated over the volume V of the body Q. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of body with the shape of the cross-section is the second moment of this area about the z-axis perpendicular to the cross-section, weighted by its density. This is also called the polar moment of the area, and is the sum of the second moments about the x and y axes.[22] The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x-axis or y-axis depending on the load.
Example calculation of moment of inertia
The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centres of mass.[18]
- The moment of inertia of a thin rod with constant cross-section s and density ρ and with length ℓ about a perpendicular axis through its centre of mass is determined by integration.[18]:1301 Align the x-axis with the rod and locate the origin its centre of mass at the centre of the rod, then
where m = ρsℓ is the mass of the rod.
- The moment of inertia of a thin disc of constant thickness s, radius R, and density ρ about an axis through its centre and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration.[18]:1301 Align the z-axis with the axis of the disc and define a volume element as dV = sr drdθ, then
where m = πR2ρs is its mass.
- The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point P as,
where L is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the centre of mass to the pivot point of the pendulum.
A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly.
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its centre of mass. This is determined by summing the moments of inertia of the thin discs that form the sphere. If the surface of the ball is defined by the equation[18]:1301
then the radius r of the disc at the cross-section z along the z-axis is
Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the z-axis,
where m = 4/3πR3ρ is the mass of the ball.
Moment of inertia in planar movement of a rigid body
If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis k perpendicular to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.[14][17][23][24]
If a system of n particles, Pi, i = 1,...,n, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point R, and absolute velocities vi
where ω is the angular velocity of the system and V is the velocity of R.
For planar movement the angular velocity vector is directed along the unit vector k which is perpendicular to the plane of movement. Introduce the unit vectors ei from the reference point R to a point ri , and the unit vector ti = k × ei so
This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.
Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.
Angular momentum in planar movement
The angular momentum vector for the planar movement of a rigid system of particles is given by[14][17]
Use the centre of mass C as the reference point so
and define the moment of inertia relative to the centre of mass IC as
then the equation for angular momentum simplifies to[18]:1028
The moment of inertia IC about an axis perpendicular to the movement of the rigid system and through the centre of mass is known as the polar moment of inertia.
For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.
Kinetic energy in planar movement
The kinetic energy of a rigid system of particles moving in the plane is given by[14][17]
This equation expands to yield three terms
Let the reference point be the centre of mass C of the system so the second term becomes zero, and introduce the moment of inertia IC so the kinetic energy is given by[18]:1084
The moment of inertia IC is the polar moment of inertia of the body.
Newton's laws for planar movement
Newton's laws for a rigid system of N particles, Pi, i = 1,..., N, can be written in terms of a resultant force and torque at a reference point R, to yield[14][17]
where ri denotes the trajectory of each particle.
The kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as,
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri and the unit vectors ti = k × ei , so
This yields the resultant torque on the system as
where ei × ei = 0, and ei × ti = k is the unit vector perpendicular to the plane for all of the particles Pi .
Use the centre of mass C as the reference point and define the moment of inertia relative to the centre of mass IC , then the equation for the resultant torque simplifies to[18]:1029
The parameter IC is the polar moment of inertia of the moving body.
The inertia matrix for spatial movement of a rigid body
The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.[3][4][5][6][25]
An important application of the inertia matrix and Newton's laws of motion is the analysis of a spinning top. This is discussed in the article on gyroscopic precession. A more detailed presentation can be found in the article on Euler's equations of motion.
Let the system of particles Pi, i = 1,..., n be located at the coordinates ri with velocities vi relative to a fixed reference frame. For a (possibly moving) reference point R, the relative positions are
and the (absolute) velocities are
where ω is the angular velocity of the system, and VR is the velocity of R.
Angular momentum
If the reference point R in the assembly, or body, is chosen as the centre of mass C, then its angular momentum takes the form,[3][6]
where the terms containing VR sum to zero by definition of the centre of mass.
To define the inertia matrix, let us first note that a skew-symmetric matrix [B] could be constructed from a vector b that performs the cross product operation, such that
This matrix [B] has the components of b = (bx, by,bz) as its elements, in the form
Now construct the skew-symmetric matrix [Δri]= [ri-C] obtained from the relative position vector Δri=ri - C, and use this skew-symmetric matrix to define,
where [IC ] defined by
is the symmetric inertia matrix of the rigid system of particles measured relative to the centre of mass C.
Kinetic energy
The kinetic energy of a rigid system of particles can be formulated in terms of the centre of mass and a matrix of mass moments of inertia of the system. Let the system of particles Pi, i = 1,...,n be located at the coordinates ri with velocities vi, then the kinetic energy is[3][6]
where Δri= ri-C is the position vector of a particle relative to the centre of mass.
This equation expands to yield three terms
The second term in this equation is zero because C is the centre of mass. Introduce the skew-symmetric matrix [Δri] so the kinetic energy becomes
Thus, the kinetic energy of the rigid system of particles is given by
where [IC] is the inertia matrix relative to the centre of mass and M is the total mass.
Resultant torque
The inertia matrix appears in the application of Newton's second law to a rigid assembly of particles. The resultant torque on this system is,[3][6]
where ai is the acceleration of the particle Pi. The kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference point, as well as the angular velocity vector ω and angular acceleration vector α of the rigid system as,
Use the centre of mass C as the reference point, and introduce the skew-symmetric matrix [Δri]=[ri-C] to represent the cross product (ri - C)x, to obtain
The calculation uses the identity
obtained from the Jacobi identity for the triple cross product as shown in the proof below:
Proof Then, the following Jacobi identity is used on the last term:
The result of applying Jacobi identity can then be continued as follows:
The final result can then be substituted to the main proof as follows:
Notice that for any vector , the following holds:
Finally, the result is used to complete the main proof as follows:
Thus, the resultant torque on the rigid system of particles is given by
where [IC] is the inertia matrix relative to the centre of mass.
Parallel axis theorem
The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the centre of mass C and the inertia matrix relative to another point R. This relationship is called the parallel axis theorem.[3][6]
Consider the inertia matrix [IR] obtained for a rigid system of particles measured relative to a reference point R, given by
Let C be the centre of mass of the rigid system, then
where d is the vector from the centre of mass C to the reference point R. Use this equation to compute the inertia matrix,
Expand this equation to obtain
The first term is the inertia matrix [IC] relative to the centre of mass. The second and third terms are zero by definition of the centre of mass C. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix [d] constructed from d.
The result is the parallel axis theorem,
where d is the vector from the centre of mass C to the reference point R.
Note on the minus sign: By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form −m[r]2, which is similar to the mr2 that appears in planar movement. However, to make this to work out correctly a minus sign is needed. This minus sign can be absorbed into the term m[r]T[r], if desired, by using the skew-symmetry property of [r].
The inertia matrix and the scalar moment of inertia around an arbitrary axis
The scalar moment of inertia, IL, of a body about a specified axis whose direction is specified by the unit vector S and passes through the body at a point R is as follows:[6]
where [IR] is the moment of inertia matrix of the system relative to the reference point R.
This is derived as follows. Let a rigid assembly of N particles, Pi, i = 1,...,N, have coordinates ri. Choose R as a reference point and compute the moment of inertia around an axis L defined by the unit vector S through the reference point R. The moment of inertia of the system around this line L=R+tS is computed by determining the perpendicular vector from this axis to the particle Pi given by
where [I] is the identity matrix and [S ST] is the outer product matrix formed from the unit vector S along the line L.
To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix [S] such that [S]y=S x y, then we have the identity
which relies on the fact that S is a unit vector.
The magnitude squared of the perpendicular vector is
The simplification of this equation uses the identity
where the dot and the cross products have been interchanged. Expand the cross products to compute
where [Δri] is the skew symmetric matrix obtained from the vector Δr=ri-R.
Thus, the moment of inertia around the line L through R in the direction S is obtained from the calculation
where [IR] is the moment of inertia matrix of the system relative to the reference point R.
This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.
The inertia tensor
The inertia matrix is often described as the inertia tensor, which consists of the same moments of inertia and products of inertia about the three coordinate axes.[6][23] The inertia tensor is constructed from the nine component tensors, (the symbol is the tensor product)
where ei, i=1,2,3 are the three orthogonal unit vectors defining the inertial frame in which the body moves. Using this basis the inertia tensor is given by
This tensor is of degree two because the component tensors are each constructed from two basis vectors. In this form the inertia tensor is also called the inertia binor.
For a rigid system of particles Pk, k = 1,...,N each of mass mk with position coordinates rk=(xk, yk, zk), the inertia tensor is given by
where E is the identity tensor
In this case, the components of the inertia tensor are given by
The inertia tensor for a continuous body is given by
where r defines the coordinates of a point in the body and ρ(r) is the mass density at that point. The integral is taken over the volume V of the body. The inertia tensor is symmetric because Iij= Iji.
Alternatively it can also be written in terms of the hat operator as:
The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction n,
where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as I12 is obtained by the computation
and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis.
The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
It is common in rigid body mechanics to use notation that explicitly identifies the x, y, and z axes, such as Ixx and Ixy, for the components of the inertia tensor.
Identities for a skew-symmetric matrix
To compute moment of inertia of a mass around an axis, the perpendicular vector from the mass to the axis is needed. If the axis L is defined by the unit vector S through the reference point R, then the perpendicular vector from the line L to the point r is given by
where [I] is the identity matrix and [S ST] is the outer product matrix formed from the unit vector S along the line L. Recall that skew-symmetric matrix [S] is constructed so that [S]y=S x y. The matrix [I-SST] in this equation subtracts the component of Δr=r-R that is parallel to S.
The previous sections show that in computing the moment of inertia matrix this operator yields a similar operator using the components of the vector Δr that is
It is helpful to keep the following identities in mind to compare the equations that define the inertia tensor and the inertia matrix.
Let [R] be the skew symmetric matrix associated with the position vector R=(x, y, z), then the product in the inertia matrix becomes
This can be viewed as another way of computing the perpendicular distance from an axis to a point, because the matrix formed by the outer product [R RT] yields the identify
where [I] is the 3x3 identity matrix.
Also notice, that
where tr denotes the sum of the diagonal elements of the outer product matrix, known as its trace.
The inertia matrix in different reference frames
The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame.[6][23] This means that as the body moves the components of the inertia matrix change with time. In contrast, the components of the inertia matrix measured in a body-fixed frame are constant.
Body frame inertia matrix
Let the body frame inertia matrix relative to the centre of mass be denoted [ICB], and define the orientation of the body frame relative to the inertial frame by the rotation matrix [A], such that,
where vectors y in the body fixed coordinate frame have coordinates x in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by
Notice that [A] changes as the body moves, while [ICB] remains constant.
Principal axes
Measured in the body frame the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix [Q] and a diagonal matrix [Λ], given by
The columns of the rotation matrix [Q] define the directions of the principal axes of the body, and the constants I1, I2 and I3 are called the principal moments of inertia. This result was first shown by J. J. Sylvester (1852), and is a form of Sylvester's law of inertia.[26][27]
For bodies with constant density an axis of rotational symmetry is a principal axis.
Inertia of an ellipsoid
The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called Poinsot's ellipsoid.[28] Let [Λ] be the inertia matrix relative to the centre of mass aligned with the principal axes, then the surface
defines an ellipsoid in the body frame. Write this equation in the form,
to see that the semi-principal diameters of this ellipsoid are given by
Let a point x on this ellipsoid be defined in terms of its magnitude and direction, x=|x|n, where n is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia In around an axis in the direction n, yields
Thus, the magnitude of a point x in the direction n on the inertia ellipsoid is
See also
- Central moment
- Instant centre of rotation
- List of moments of inertia
- Rotational energy
- Stretch rule
- Tire balance
- 1 2 Mach, Ernst (1919). The Science of Mechanics. pp. 173–187. Retrieved November 21, 2014.
- ↑ Euler, Leonhard (1765). Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all motions which can occur in such bodies] (in Latin). Rostock and Greifswald (Germany): A. F. Röse. p. 166. ISBN 978-1-4297-4281-8. From page 166: "Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur." (Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)
- 1 2 3 4 5 6 Marion, JB; Thornton, ST (1995). Classical dynamics of particles & systems (4th ed.). Thomson. ISBN 0-03-097302-3.
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- ↑ Wolfram, Stephen (2014). "Spinning Ice Skater". Wolfram Demonstrations Project. Mathematica, Inc. Retrieved September 30, 2014.
- ↑ Hokin, Samuel (2014). "Figure Skating Spins". The Physics of Everyday Stuff. Retrieved September 30, 2014.
- ↑ Breithaupt, Jim (2000). New Understanding Physics for Advanced Level. Nelson Thomas. p. 64. ISBN 0748743146.
- ↑ Crowell, Benjamin (2003). Conservation Laws. Light and Matter. p. 107. ISBN 0970467028.
- ↑ Tipler, Paul A. (1999). Physics for Scientists and Engineers, Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics. Macmillan. p. 304. ISBN 1572594918.
- 1 2 3 4 5 Paul, Burton (June 1979). Kinematics and Dynamics of Planar Machinery. Prentice Hall. ISBN 978-0135160626.
- ↑ Walker, David Halliday, Robert Resnick, Jearl (2005). Fundamentals of physics (7th ed.). Hoboken, NJ: Wiley. ISBN 9780471216438.
- ↑ French, A.P. (1971). Vibrations and waves. Boca Raton, FL: CRC Press. ISBN 9780748744473.
- 1 2 3 4 5 6 Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2010). Theory of Machines and Mechanisms (4th ed.). Oxford University Press. ISBN 978-0195371239.
- 1 2 3 4 5 6 7 8 9 10 Ferdinand P. Beer; E. Russell Johnston; Jr., Phillip J. Cornwell (2010). Vector mechanics for engineers: Dynamics (9th ed.). Boston: McGraw-Hill. ISBN 978-0077295493.
- ↑ H. Williams, Measuring the inertia tensor, presented at the IMA Mathematics 2007 Conference.
- ↑ Gracey, William, The experimental determination of the moments of inertia of airplanes by a simplified compound-pendulum method, NACA Technical Note No. 1629, 1948
- ↑ In that situation this moment of inertia only describes how a torque applied along that axis causes a rotation about that axis. But, torques not aligned along a principal axis will also cause rotations about other axes.
- ↑ Walter D. Pilkey, Analysis and Design of Elastic Beams: Computational Methods, John Wiley, 2002.
- 1 2 3 Goldstein, H. (1980). Classical Mechanics (2nd ed.). Addison-Wesley. ISBN 0-201-02918-9.
- ↑ L. D. Landau and E. M. Lifshitz, Mechanics, Vol 1. 2nd Ed., Pergamon Press, 1969.
- ↑ L. W. Tsai, Robot Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999.
- ↑ Sylvester, J J (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares" (PDF). Philosophical Magazine (Ser. 4). 4 (23): 138–142. doi:10.1080/14786445208647087. Retrieved June 27, 2008.
- ↑ Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 0-19-853248-2.
- ↑ Mason, Matthew T. (2001). Mechanics of Robotics Manipulation. MIT Press. ISBN 978-0-262-13396-8. Retrieved November 21, 2014.
External links
Wikimedia Commons has media related to Moments of inertia. |
- Angular momentum and rigid-body rotation in two and three dimensions
- Lecture notes on rigid-body rotation and moments of inertia
- The moment of inertia tensor
- An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)
- Tutorial on finding moments of inertia, with problems and solutions on various basic shapes
- Notes on mechanics of manipulation: the angular inertia tensor


text: On Tue, Dec 07, 2004 at 02:51:24PM -0800, pesch at attglobal.net wrote: > Where can I find a decent comparison betwenn Opteron and Athlon64 Unfortunately not much Linux-relevant performance info is published about Athlon64; nobody publishes SPECcpu numbers, for example. Lots of people publish Windows desktop performance info for Athlon64, but often Windows people don't publish Opteron numbers. So... -- greg

subset: commoncrawl


text:
Deletes a set of rows from a table or view.
Embedded, Interactive, Module, ODBC, Procedural, JDBC.
All rows in the set defined by the WHERE clause are deleted from the base table or view identified by table-name. If no WHERE clause is specified, all rows are deleted.
If table-name identifies a view rather than a base table, entire rows, including columns invisible from the view, are deleted from the base table on which the view is defined. For a delete to be performed on a view, the view must be updatable, see CREATE VIEW.
A NOT FOUND condition code is returned if no row is deleted, see Return Status and Conditions.
Language Elements
search-condition, see Search Conditions.
with-clause, see The WITH Clause.
DELETE privilege must be held on the table or view identified by table-name.
In a procedural usage context, the DELETE statement is only permitted if the procedure access-clause is MODIFIES SQL DATA, see CREATE PROCEDURE.
If a correlation-name is specified after the table name in the DELETE FROM clause, the correlation name must be used to refer to the table in the WHERE clause of the DELETE statement.
If the table name addressed by the DELETE statement is subject to any referential constraints, the delete operation must not create a situation where these constraints are violated. The effect of the delete operation on any referential constraints depends on the delete-rule in effect for each constraint, see CREATE TABLE for a description of the delete-rule options.
A DELETE statement is executed as a single statement. If an error occurs at any point during the execution, none of the rows in the set defined by the WHERE clause will be deleted (however, if the table is stored in a databank with the WORK option it is possible that some rows will be deleted).
DELETE FROM countries
WHERE CITY = 'Dublin';
Standard Compliance
Standard |
Compliance |
Comments |
Core |
Fully compliant. |


text: The development of Black music in the United Kingdom owes much to the influence of Jamaican and US music on each succesive generation of Carribean immigrants to the UK - from Windrush to the present day. UK based MC's take these influences, mix them up with local references and styles thus creating an identity of Black British culture. This is constantly evolving not just musically but also in terms of each subsequent generation's relationship to British society. As Tricky put it in Massive Attack's Blue Lines, of "English upbringing, background Caribbean" An England Story shows the links and musical path from the arrival of UK Dancehall and Soundsystems in the early 1980s, through successive musical movements such as Jungle, UK Hip-Hop, up to today's Garage, Grime and Dubstep. The global pre-eminence of American hip hop means that music like grime and UK hip hop is often seen as a form of rap, whereas it owes as much to reggae music and culture as it does to any American influence. Black music in Britain has fashioned its own identity in contrast to that of America, Africa, or elsewhere, by drawing on the unique relationship that the UK has with the Caribbean. As Rodney P (of celebrated 80s UK Hip-Hop group London Posse) says: "This is a UK thing, it's hip hop and it's reggae and we do reggae - and those Americans don't know about that". Since the evolution of the 'fast chat' dancehall style of the early 80s, the influence of reggae music and culture has been crucial to the development of urban music in Britain: from the heavy sampling of Jamaican vocals/instrumentals employed by jungle, hardcore and garage, to grime's London take on the soundclash, the riddim version and patois-inflected rhymes. The importance of the Jamaican soundsystem concept and its dubplates, specials, clashes and heavy, heavy bass is a constant throughout these different stylistic mutations "An essential primer for your head, hips and feet." The Wire "Two-disc CD celebrating the history of the MC. Crank this up!" DJ "Soul Jazz records' excellent new compilation says more about cultural integration than any number of Mobo awards" The Guardian Read the full review Here "England Story is a vital reminder of the power of the voice. Essential" Observer Music Monthly. Read the Full Review Here "The electrifying creativity on An England Story hints at how great the future could be. Bigots of all colours will hate it." New Statesman. Read the full review Here "As much a history lesson as a compilation" Q Please note: Tracklist for LP Vol.1 is tracks 1-11 Tracklist for LP Vol.2 is tracks 12-21


text:
164 Nautica Mile Drive
Clermont, Florida, 34711
Commute to Downtown Clermont
Almost all errands require a car.
Minimal bike infrastructure.
164 Nautica Mile Drive has a Walk Score of 20 out of 100. This location is a Car-Dependent neighborhood so almost all errands require a car.
This location is in Clermont. Nearby parks include Clarence Bishop Memorial Field and William McKinley Park.
Explore how far you can travel by car, bus, bike and foot from 164 Nautica Mile Drive.
This location is in the city of Clermont, FL. Clermont has an average Walk Score of 21 and has 28,742 residents.
Learn More About Clermont


text:
Free forum : Hamradio and electronics help and advice for visitors to Harry's Homebrew Homepage (www. sm0vpo. com). Also a peek at tubes, nostalgia and radio as it used to be. Remember the red spot and white spot transistors? Philips EE? Trionic? I a
sm0vpo, projects, hamradio, electronics, advice, help, harry's, homebrew, homepage, radio, #transmitter, receiver, valves, tubes, synthesiser, synthesizer, vintage, memory, lane
U Can Have All Transmitted Circuits Here
#transmitter, circuits, have, transmitted, here


text:
As the conversion to digital projection continues to transform cinema exhibition, much of the enduring imagery of movies — reels, film, a booth, a projectionist — is fast becoming obsolete.
That's one of the reasons that Boston filmmaker Peter Flynn is making a documentary about cinema projection, traveling the country to big-city and small neighborhood theaters, visiting booths and the projectionists inside them. His film, "The Dying of the Light," endeavors to, as he puts it, "capture the craft of film projection at its moment of eclipse."
Flynn, who teaches film at Emerson College and for 10 years ran the Boston Irish Film Festival, recently launched a website (www.dyingofthelight
film.com) to archive the footage he's shot, and to promote his film. He plans to keep updating the site as a repository for clips, photos, and information, even after his film is finished.
Flynn, who earned a master's in film studies from University College of Dublin, arrived in Boston in 1994. He directed his first feature documentary last year, "Blazing the Trail: The O'Kalems in Ireland," about Sidney Olcott and Gene Gauntier of New York's Kalem Film Co., who made a series of films in Ireland in the early 1910s. He set out one year ago to make a documentary about art cinemas, he says, but quickly discovered that he was more interested in the fast-disappearing culture of projectionists and the cinema history contained in the booths. As Rick Shamel, who has a projection room in his Acton home, says in the film, "The equipment is a connection to the past. A projectionist from the 1920s could come in here and run this. Nothing has changed."
Flynn has shot interviews with projectionists Dave Leamon (Brattle Theatre, Cambridge), Matt Gress (Coolidge Corner Theatre, Brookline), Bob Throop (Capitol Theatre, Rome, N.Y.), Jason Sheckley (Harbor Theatre, Boothbay Harbor, Maine), and Raber Umphenour (Avon Cinema, Providence). This week, he heads to Indianapolis, then to New York to see the former head projectionist at Radio City Music Hall. "They've all been so generous in sharing stories," says Flynn. "Once I started listening to them, I knew I wanted to capture all this stuff and get it out there as quickly as possible. Most audiences don't really know how film is projected, or what goes on in that booth that's right behind their backs."
For more information, go to www.dyingofthelightfilm.com.
Digital revolution
On the subject of digital conversion, the Coolidge Corner Theatre in Brookline just announced that it has successfully raised all of the $223,000 needed to upgrade its main screens, thanks to grants and donations.
"The transition to digital projection is arguably the biggest transformation to hit the film industry since the advent of sound, and unfortunately, it may spell the end for many small cinemas that cannot afford this upgrade. As such, I cannot possibly overstate how grateful we are to be a part of such a generous community, and to have audiences who recognize the value of the innovative, independent film programming we offer," said Denise Kasell, executive director of the Coolidge Corner Theatre Foundation, in a press release.
The Brattle Theatre in Harvard Square, meanwhile, is halfway to its $140,000 fund-raising goal, with a deadline of Feb. 28, largely as a result of using the Kickstarter crowdsourcing website. The Brattle campaign aims to finance two projects: a digital projection system and an HVAC upgrade. "Distributors are now sending hard drives to theaters, rather than the familiar reels of 35mm film," said Ned Hinkle, the Brattle's creative director, on Kickstarter. "To be clear, we'll still show 35mm prints after the upgrade. But we need DCP [Digital Cinema Package] in order to continue our one-of-a-kind repertory programming, because many movies will only be available digitally going forward." One thing that won't change: the Brattle's rear-projection system, thought to be the only one still operating in the country.
Go to www.kickstarter
Art on film
Cape Cod resident Richard Miller knew he was witnessing something special back in 1963 when he was working at the Munson-Williams-Proctor Institute in Utica, N.Y. The museum hosted the 50th anniversary re-creation of the 1913 International Exhibition of Modern Art, which took place at the 69th Regiment Armory in Manhattan. Known thereafter as the Armory Show, it's considered a seminal moment in the history of modern art in America. Artist Marcel Duchamp, then 75, delivered a lecture at the Utica museum about the Armory Show and its impact on the development of modern art. Miller recorded the lecture and preserved it for all these years.
Two years ago, he began efforts to turn his recording into a film. His research included licensing images for the many famous works of art showcased in the Armory Show. "Marcel Duchamp and the Show That Shook the World," Miller's 60-minute film, will have its debut screening at the Provincetown Art Association and Museum Sunday at 2:30 p.m. It's also slated to screen at the Cultural Center of Cape Cod in Yarmouth on April 5.
For more information, go to www.paam.org.
Loren King can be reached at email@example.com.


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Rick Renzi on Social SecurityRepublican Representative (AZ-1) |
The mission of the Alliance for Retired Americans is to ensure social and economic justice and full civil rights for all citizens so that they may enjoy lives of dignity, personal and family fulfillment and security. The Alliance believes that all older and retired persons have a responsibility to strive to create a society that incorporates these goals and rights and that retirement provides them with opportunities to pursue new and expanded activities with their unions, civic organizations and their communities.
The following ratings are based on the votes the organization considered most important; the numbers reflect the percentage of time the representative voted the organization's preferred position.
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Other candidates on Social Security: | Rick Renzi on other issues: | |||
AZ Gubernatorial: Jan Brewer Janet Napolitano-2008 AZ Senatorial: John McCain Jon Kyl Pending elections: D,IL-5:Emanuel D,CA-31:Solis D,NY-20:Gillibrand Special elections in 110th Congress: R,GA-10:Broun D,IN-7:Carson D,LA-6:Cazayoux D,MD-4:Edwards D,IL-14:Foster D,CA-37:Richardson R,LA-1:Scalise D,CA-12:Speier D,MA-5:Tsongas R,VA-1:Wittman |
GOP Freshmen in 111th Congress: R,OH-7:Austria R,OH-16:Boccieri R,LA-2:Cao R,LA-6:Cassidy R,UT-3:Chaffetz R,CA-6:Coffman R,LA-4:Fleming R,KY-2:Guthrie R,MS-3:Harper R,CA-52:Hunter R,KS-2:Jenkins R,NJ-7:Lance R,NY-26:Lee R,MO-9:Luetkemeyer R,WY-AL:Lummis R,CA-4:McClintock R,TX-22:Olson R,MN-3:Paulsen R,FL-15:Posey R,TN-1:Roe R,FL-16:Rooney R,IL-18:Schock R,PA-5:Thompson |
Dem. Freshmen in 111th Congress: D,NJ-3:Adler D,AL-2:Bright D,VA-11:Connolly D,PA-3:Dahlkemper D,OH-1:Driehaus D,OH-11:Fudge D,FL-8:Grayson D,AL-5:Griffith D,IL-11:Halvorson D,NM-1:Heinrich D,CT-4:Himes D,OH-15:Kilroy D,AZ-1:Kirkpatrick D,NC-8:Kissell D,FL-24:Kosmas D,MD-1:Kratovil D,NM-3:Lujan D,NY-25:Maffei D,CO-4:Markey D,NY-29:Massa D,NY-13:McMahon D,ID-1:Minnick D,VA-5:Perriello D,MI-9:Peters D,ME-1:Pingree D,CO-2:Polis D,MI-7:Schauer D,OR-5:Schrader D,NM-2:Teague D,NV-3:Titus D,NY-21:Tonko |
Budget/Economy Civil Rights Corporations Crime Drugs Education Energy/Oil Environment Families/Children Foreign Policy Free Trade Govt. Reform Gun Control Health Care Homeland Security Immigration Infrastructure/Technology Jobs Principles/Values Social Security Tax Reform War/Iraq/Mideast Welfare/Poverty Main Page Profile AZ politicians |


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Get ready for a real militarization in Ferguson, Missouri. After two nights of unrest following a softer approach by the state highway patrol, Governor Jay Nixon has ordered the National Guard to deploy in Ferguson to enforce a midnight curfew. Nixon blamed outside agitators for the continuing confrontations in the street after the shooting of Michael Brown, an unarmed black teenager suspected of taking part in a strongarm robbery just before the fatal confrontation:
Missouri's governor said on Monday he would send the National Guard into the St. Louis suburb of Ferguson to restore calm after authorities forcibly dispersed a crowd protesting last week's fatal shooting of an unarmed black teen by police.
Gov. Jay Nixon signed an executive order deploying the U.S. state militia, saying demonstrators had thrown Molotov cocktails and shot at police as well as a civilian, a description of the night's events diverging widely from some eyewitness accounts.
"Tonight, a day of hope, prayers, and peaceful protests was marred by the violent criminal acts of an organized and growing number of individuals, many from outside the community and state, whose actions are putting the residents and businesses of Ferguson at risk," Nixon said in a statement on his website.
The commander of the state highway patrol, Captain Ron Johnson, went from hero to enemy in a few short hours after ordering action against would-be rioters before the curfew went into effect:
In a local church, people gave a standing ovation on Sunday afternoon to Capt. Ron Johnson of the Missouri Highway Patrol.
But hours later on W. Florissant Ave. — near where Officer Darren Wilson killed unarmed Michael Brown with six bullets — Johnson was seen as the enemy. …
The people of Ferguson rose to their feet and exploded in applause. They needed a hero.
But by 11 p.m., and a mile away, a small band of protesters again clashed with Capt. Johnson's highway cops.
Police claim they were assaulted. They responded with tear gas.
Johnson explained why police, after falling back the previous night and allowing rioters to go on a rampage, responded with teargas and a more aggressive strategy last night and this morning. In part, Johnson insisted, it had to do with police forces defending themselves from attack by Molotov cocktails, as well as protecting the public and property:
Sunday night marked the second evening of a state-imposed midnight-5 a.m. curfew Nixon put in place, a decision he said was necessary to prevent looters from doing more damage but has been criticized by many in the Ferguson community and elsewhere.
Police officials have acknowledged that they fired several smoke canisters and at least one tear gas canister Sunday, and many of the hundreds of officers in Ferguson Sunday evening appeared in riot gear. Protesters said the police acted without being provoked, while the police reported that they were responding to gunfire and Molotov cocktails thrown by members of the crowd. Missouri State Highway Patrol Capt. Ron Johnson said one protester shot another and that the victim was listed in critical condition early Monday morning. He also said he was forced to "elevate the level" of police response after some crowd members threw bottles at officers.
In other Ferguson news, the Brown family released the results of a private autopsy conducted on a pro bono basis by Michael Baden. Brown was hit by at least six bullets in the shooting — but none of the bullets struck him from behind, as his friend claimed:
#MichaelBrown's family releases private autopsy results. http://t.co/DEJNzkTBQ1 #5Things #NewDay pic.twitter.com/1k5FtIvyQx
— New Day (@NewDay) August 18, 2014
"There were at least six entry wounds, there might have been seven, but we'll have to correlate that with what was found in the first autopsy," Baden, who retired from the New York state police in 2011, told the Wall Street Journal.
He cautioned against drawing conclusions from the autopsy. "Right now there is too little information to forensically reconstruct the shooting," he told the Times, but added: "In my capacity as the forensic examiner for the New York State Police, I would say, 'You're not supposed to shoot so many times.'"
He also told the Times about Brown's head wounds.
"This one here looks like his head was bent downward," he said, indicating a wound at the top of Brown's head. "It can be because he's giving up, or because he's charging forward at the officer."
Baden, 80, a veteran medical examiner who has performed 20,000 autopsies, reviewed the autopsies of John F. Kennedy and Martin Luther King Jr. and hosted the HBO show "Autopsy," said the bullets hit Brown in the front, but the absence of gunpowder showed they were not fired at close range. Baden also said nothing indicated Brown had been in an altercation. This seems to contradict the statements of Brown's friend, Dorian Johnson, who said the officer now identified as Darren Wilson grabbed Brown's neck with one hand and shot him with the other.
So do the direction of the bullets. Johnson had claimed that Brown was running away with his hands up when he was shot. The hands-up posture has been adopted by protestors in Ferguson as a rebuke to the police for the shooting. However, the autopsy finding about the shot to the arm appears to contradict that part of Johnson's statement as well:
The teen was struck once in the top of head, once in the forehead and four times in the arm, Dr. Baden said. Some of the shots to his arm went through the limb and entered his chest and lungs, according to Dr. Baden, who served for 25 years in the medical examiner's office in New York City and another 25 years with the New York State Police before entering private practice.
If Brown's arms were up when he was shot, then Johnson's claim seems difficult to reconcile with this evidence, although Brown's arms could have come down between shots, too. The lack of gunpowder suggests that Brown was at least some distance from Officer Darren Wilson when he was shot, though, which intensifies the question about the use of lethal force, given that Brown was unarmed at the time.
However, at least in Baden's autopsy — performed on behalf of Brown's family — Wilson didn't shoot Brown as he fled, as Johnson claimed. That claim played a large part in stoking the rage that led to the riots. Perhaps the Department of Justice thinks it will find something more than Baden did in his autopsy, but Baden's report and its provenance will be difficult to refute. Maybe the National Guard can calm the situation long enough for everyone to catch their breath and catch up to the facts in the case.


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Dear Editor:
What a fabulous music show I saw last night at Lincoln Harbor Park on the Weehawken waterfront as part of the Hudson Riverfront PAC's Summer Concerts on the Hudson series. And it was free! I brought my blanket (though chairs were provided), and a picnic to see Mountain Heart, a fantastic 6-piece Country Rock, Blues & Bluegrass band from Nashville, Tennessee, of whom I'd never heard. They performed their own songs as well as some bluesy classics that had the audience enthralled and, in some cases, up and dancing. A few friends stopped by my blanket who had never been to one of these concerts before and I was amazed. In 8 years of these concerts, I've hardly missed any. I often don't know the bands or groups but I'm always impressed and always have a great time as we did last night. The weather was beautiful (after being postponed one night because of hail storms the night before!); the players were fine musicians, fun and inspiring; and there was not a better backdrop to any stage. All for free! When the band closed with a rendition of Billy Joel's "New York State of Mind" on piano & fiddle with the NYC skyline in the background it was the end of a perfect evening—and did I mention it was free!
Why weren't more people in attendance? Billboards for the series are everywhere around town; there's always an article in the Weehawken Reporter about the events, and the Library and other establishments give away a handy card with info that I keep on my fridge. Summer Concerts on the Hudson offers some of the best entertainment this side of the Hudson. Come on out, people! Bring your neighbors! It's all for us and it's all free! Bring a blanket and a picnic and you'll be rewarded with some of the finest sounds and sights that Weehawken and Hudson County has to offer.
Donna Hochhauser
actor/musician/Weehawken resident


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Americans off to finals
Americans off to finals
Beat Finns 3-0 after scoreless 53 minutes
The goal came as a result of a nice passing play with Gigi Marvin and Amanda Kessel, Knight finishing the play with a spin in the slot and a backhand that just squirted through the pads of goalie Noora Raty, who was sensational in the Finnish goal.
"It's always nice to score a goal when your team needs it most," Knight said. "We're happy that got the tide rolling and we were able to put in a couple more."
Monique Lamoureux added a power-play goal two minutes later, ending the possibility of a Finnish "ihme" (miracle) on ice. Brianna Decker added an empty netter to finish the scoring.
Jessie Vetter stopped just 11 shots for the shutout. Raty faced 48 U.S. shots.
The Americans dominated the first 40 minutes in a way that suggested a lead of 5-0 or 6-0 would not be unexpected. There was only one reason this wasn't the score line—Raty.
"We always believed with persistence we could get one by her," Amanda Kessel said of Raty. "We knew we had to stay positive and just keep shooting."
In the first period Raty was deked down and out by a patient Jocelyne Lamoureux but made a lunging glove save, and she later made a nice series of saves after some fine passing between Lyndsey Fry, Alex Carpenter, and Anne Schleper.
"We knew it would be tough to score against "Ratsy"," Knight said. "She's a great goaltender. "At the same time, we knew we'd put the puck in the net. It was just a matter of time."
Not so fast.
The second period was dominated by penalties, but the Finns not only killed off two situations of three-on-five, they came dangerously close to opening the scoring. With about seven minutes left in the second, and teams playing four aside, Michelle Karvinen undressed defenceman Gigi Marvin and ripped a wicked shot off the crossbar that stayed out. Video review confirmed the closeness of the play, but the call of "no goal" was the correct one.
Continue readingRaty continued to be sensational but saved her best for last. With just 3.2 seconds left in the middle period she robbed Hilary Knight from in close, and the frustrated Americans had a little pushing and shoving session with the Finns at the whistle which resulted in a Finnish power play.
Shots in the second were 24-3 (34-8 over two periods), but the score was 0-0. The Finns were short-handed 9:24 through the first two periods compared to just 2:00 for the Americans, but the Suomi penalty killers, led by Raty, were nothing short of sensational.
That set up a most interesting third period, but after nearly 14 more minutes without a goal, Knight struck a dagger into Finland's gold-medal game hopes.
"We didn't have any pressure on us. It was all on them," Finland's Michelle Karvinen noted. "We had a really good feeling going out for the third period, but things just didn't go our way. We have one more game left, so we're going to give everything we have tio get a medal."
Back to Overview


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Maxell 315 SR716SW
Regular price
Regular price
Sale price
Unit price
Maxell Silver Oxide Watch Battery SR716SW Low Drain Replaces 315 SR67 The 315, or SR716SW, is a Silver-Oxide cell normally used in watches, with a shelf life of 5+ years. Other applications may include calculators, film cameras, medical instruments, cash registers, or FA instruments (measuring instruments, onboard microcomputers, sensors). This is a "SW" low drain battery. Low drain batteries are best used in devices that use a small amount of power over a long period of time: Analog watches, TV remotes, alarm clocks, etc. This is in comparison to high drain batteries, which are for devices with a continuous, low power use and occasional high peaks: Analog and digital watches with extra functions (backlight, alarm), pocket calculators, electronic games, cameras, etc. Specifications Chemistry: Silver Oxide Voltage: 1.55V Capacity: 22mAh Diameter: 7.9 mm Height: 1.6 mm Features Low Drain Type – Best used with analog watches A flat discharge curve during discharge supplies a stable voltage until the end of the discharge life High-energy density per unit volume provides 2x the amount of energy capacity as button-type alkaline batteries Featuring Maxell's original leak-resistant processing Also Known As: 1187SO, 280-56, 314, 315, 530, 614, B315L, D 315, D315, GP315, HA, R 315/40, RW 316, SB-AT, SP315, SR 62, SR 716 SW, SR67, SR716, SR716SW, SR716W, V314, V315 Need more than one? We offer great discounts on quantity orders and FREE SHIPPING offers available. Specifications Brand: Maxell Output Voltage: 1.55V Amperage: 22mAh Chemistry: Silver Oxide Battery Size: 315 / 314 Height: 7.9mm Diameter: 1.6mm Country of Manufacture: Japan Compatibility Maxell: SR716SW Size: 1187SO, 280-56, 314, 315, 40, 530, 614, B315L, D315, GP315, HA, R315, RW316, SB-AT, SP315, SR62, SR67, SR716, SR716SW, SR716W, V314, V315


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Create a Website Account - Manage notification subscriptions, save form progress and more.
If your pet is impounded and you live in the City you must first purchase a license for your pet which means the rabies vaccination must also be current. Once this is taken care of, you must pay the impound fee and the animal shelter maintenance costs. The current impound fees are: 1st offense, $25, 2nd offense, $50 and 3rd offense $75 (in 6 months)
Show All Answers
The City does not issue pet licenses. Owners are required to have proof of rabies vaccination as well as identification on their pet or be microchipped.
The ordinance states that a structure can house no more than three animals (of any combination of cats and/or dogs).
Yes. Pets can not run at large and must either be on a leash under the control of a person of suitable age, be securely tethered in their own yard or be effectively confined within an enclosure, building or motor vehicle.
Owners must clean up after their pets on all public and private property. Pet owners are also not allowed to let excrement accumulate in their own yards for more than 7 days.
Yes. It is legal for any person to seize an animal running at large. You are required to report that you have the animal within six hours to the police. If you know who the animal belongs to, you are required to give this information to the police.
Call the Buffalo Police Department at 682-5976 (after hours call 682-1162) Records are kept of all impounded animals. After 5 days, records are maintained at the animal shelter where pets are impounded. Animals are held for five business days and then put up for adoption. Animals that are not claimed or adopted could be destroyed.
This should be reported to the Buffalo Police Department by dialing 911 or 682-1162. If the injury is serious enough to require medical attention, any treating physician is required to file a report with Chief of Police.The incident will be investigated and the animal in question will have to be quarantined for a period of 10 days. To be quarantined the animal must be kept inside the home or at a veterinary hospital at the owner's expense.
If the attack is unprovoked, the animal could be deemed: "potentially dangerous." The owner is notified of this determination in writing. If the animal attacks someone again, it must be removed from the City immediately.
It is against the City Ordinance to allow pets to bark or whine excessively. Often the reason these pets are barking or whining is that they have been left unattended. Pet owners should be aware that pets left alone for long periods of time have a greater tendency to bark. If you are going to be out of town, make sure someone is taking proper care of your pets while you are gone. Call the Police Department at 682-1162 to report violators.
No, wild animals are not allowed in the city limits.


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ALL MY CHILDREN Cast: Top row (L-R) - Heather Roop, Jordi Vilasuso, Darnell Williams, Eric Nelsen, Cady McClain, Vincent Irizarry, Denyse Tontz, Sal Stowers, Robert Scott Wilson, Jordan Lane Price; Middle row (L-R) - Ray MacDonnell, Francesca James, Debbi Morgan, Jill Larson, David Canary, Ryan Bittle; Bottom row (L-R) - Lindsay Hartley, Julia Barr, Eden Riegel; Photo Credit: Chapman Baehler |
ALL MY CHILDREN premieres with all new episodes on Hulu, Hulu Plus and iTunes on Monday, April 29th.
Check out day-by-day spoilers for the first three weeks of episodes below:
Monday, 4/29:
A Pine Valley resident vows vengeance. Jesse plans a surprise. As a fast car takes a son home, Pete spots something that will change everything.
Tuesday, 4/30:
A Pine Valley legend returns. Who learns something has gone terribly wrong? Dixie sits vigil.
Wednesday, 5/1:
An ex-con learns who helped him. Jesse gets big news. Cara fears the truth will come out.
Thursday, 5/2:
AJ helps Miranda out of a tight situation. After a year and half, we finally learn the answer to an age-old question: Which character dies today?
Friday, 5/3: On MORE ALL MY CHILDREN, the behind-the-scenes show, interact with your favorite AMC stars as they discuss this week's hot topics and take a closer look at the big events in Pine Valley from the past four days.
Monday, 5/6: A rebellious side comes out, leading to an exciting adventure. Jesse's worst nightmare is confirmed. Who will rejoin the living?
Tuesday, 5/7: Miranda's feelings catch Bianca by surprise. Who will awake handcuffed to a bed? Celia can't believe what she overhears.
Wednesday,5/8: Who will bid farewell to Pine Valley? The truth sets someone free—almost. One mom gets an early Mother's Day surprise.
Hot Chelle Rae performs for Denyse Tontz (Miranda) and Eric Nelsen (AJ) |
Friday, 5/10: Friday recap show.
Monday, 5/13: Angie learns the unthinkable. How will David take the news?
Tuesday, 5/14: Who is Sinnamon? Miranda lends a hand to a friend.
Wednesday, 5/15: Opal does what she does best—meddles. David plots a reinvention.
Thursday, 5/16: Pete sets a sophisticated plan in motion. Just when you thought it was safe …a fan-favorite returns. (And does another one finally die?)
Friday, 5/17: Friday recap show.
Brand new, 30-minute episodes of ALL MY CHILDREN will be available to stream online weekdays in HD (when available) via the free Hulu (www.hulu.com) service and to the millions of Hulu Plus subscribers watching on connected TVs, mobile phones, tables and PCs. In addition, the iTunes Store will offer both series via iPhone, iPad, iPod Touch, Apple TV and Mac and PC, making these venerable dramas available in a new format that suits the viewing habits of the digital generation.
- ALL MY CHILDREN Premiere Red Carpet Interviews On WE LOVE SOAPS TV
- FX Canada Will Air ALL MY CHILDREN & ONE LIFE TO LIVE Beginning April 29
- INTERVIEW: Getting To Know ALL MY CHILDREN's Denyse Tontz
- Prospect Park/TOLN News Hub


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Statement of Eileen Claussen
President, Center for Climate and Energy Solutions
June 25, 2013
President Obama is laying out a credible, comprehensive strategy to use the tools at his disposal to strengthen the U.S. response to climate change. His plan recognizes that the costs of climate change are real and rising, and that to minimize them we must both cut our carbon output and strengthen our climate resilience. Putting these critical issues before the American public is itself a step forward. But it will require continued presidential leadership to translate the plan's good intentions into concrete policy.
The most cost-effective way to reduce greenhouse gas emissions is for Congress to enact an economy-wide price on carbon. As long as Congress is unwilling to act, the president is right to use his powers under the Clean Air Act to curb emissions from power plants, by far the largest unregulated source of U.S. carbon emissions. Many companies are prepared to work with the administration on pragmatic approaches that cut emissions while keeping U.S. electricity affordable and reliable. Companies want regulatory certainty and know that continued inaction exposes them to increasing climate risks.
In crafting the power plant rules, EPA should consult widely with utilities and with the states, which ultimately must implement them. We strongly encourage EPA to devise a flexible strategy that allows a variety of state-level policies, including market-based approaches, and allows utilities to cut emissions at the lowest possible cost.
With extreme weather and other climate impacts being felt across the country, the president also is right to place equal emphasis on strengthening America's climate resilience. His plan will help communities and businesses apply the lessons learned from Hurricane Sandy and other extreme events to better cope with future climate risks.
Implementing the president's plan will be extremely challenging. But a clear majority of the American public favors stronger climate action, and with a plan in place, the administration must now follow through with a true sense of urgency. We look forward to working with the administration as it moves forward.
Contact: Laura Rehrmann, 703-516-4146, email@example.com
About C2ES
The Center for Climate and Energy Solutions (C2ES) is an independent, nonprofit, nonpartisan organization promoting strong policy and action to address the twin challenges of energy and climate change. Launched in November 2011, C2ES is the successor to the Pew Center on Global Climate Change. Learn more at www.c2es.org.


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Whatever your reasons may be, if you're going to be holding a camera for long periods of time this shoulder mount will both steady the image, and help save you some aches and pains. [Kyle Jason] built the rig seen above for just $20 by following this guide.
[Knoptop] published the guide about a year ago. It doesn't make use of any special PVC connectors, so you'll have no problem finding everything you need at the hardware store. Connectors used include 45 and 90 degree angles, straight pieces, and a PVC conduit box to serve as the mounting bracket. After cutting, dry fitting, and welding everything together the rig really benefits from a couple of coats of paint. Don't forget the grip covers to make the thing easy to hold onto.
Don't want to read the build guide? After the break you'll find [Knoptop's] build video which is actually quite a fun eight minutes to watch.
[via Reddit]


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Where's All the Stuff? Almost Naked Kitchen Walls
This goes beyond open storage. We're talking completely cabinet-less walls here, with room for art instead of canned goods. What do you think? Modern and clean or too un-kitchen-like?
Of course, you need enough storage elsewhere to keep your walls this bare. There need to be a lot of bottom cabinets or a butler's pantry, for heaven's sake. We're guessing not many of our Small, Cool contestants have the luxury of leaving their walls naked.
On the other hand, all of these spaces feel so peaceful. We'd love to look up from our chopping and see a great piece of art. And we've bumped our heads so many times on open cabinet doors that it would feel nice to move around a room with so much open, blank space.
Even the open shelves in the last photo look more like a staging area for objets d'art than for dishes.
• 1. Completely bare and stark white, from Living, Etc.
• 2. Leaning modern art, from Dwell. (We suspect there may be cabinets to the left, but we still love this big, open wall.)
• 3. Colorful art in a row, from Southern Living.
• 4. Cool concrete, from Canadian House & Home.
• 5. Meticulously styled shelves, from Coastal Living.
Which one is your favorite?
Related: Modern to Traditional: What Kind of Kitchen are You?
(Images: Jake Curtis/Living, Etc.; Roger D'Souza/Dwell; Tria Giovan/Southern Living; Ted Yarwood/Canadian House & Home; Roger Davies/Coastal Living)