= Chapter 6

== Uniform Circular Motion

Speed is constant but not velocity, because direction is constantly changing.

Centripetal acceleration for uniform circular motion: $ a = v^2/r = ((2 pi)/T)^2 r$

*Period:* Time taken to go around circle one time.

*Frequency:* Number of revolutions per second: $ f = 1/T$ (unit is $upright(s^(-1))$)

*Angular velocity:* $omega = 2 pi f$

*Speed:* Time taken to make one revolution: $v = (2 pi r)/T = (2 pi f r) = r omega$

== Dynamics of Uniform Circular Motion

Net force producing the centripetal acceleration of uniform circular motion: $hat(F)_"net" = m hat(a)$ (towards the center of the circle)

When a car turns in an _unbanked_ (horizontal/level) circle, static friction is the force that causes centripetal acceleration.

When a car turns in a _banked_ circle, normal force is the force that causes centripetal acceleration instead.

== Apparent Forces in Circular Motion}

Apparent weight equals normal force.

*Critical speed:* $v_c$ is the speed for which $hat(n) = 0$. \
For roller coaster, $v_c = sqrt(g r)$ \
The critical speed is the slowest speed at which the car can complete a roller coaster circle.

== Circular Orbits and Weightlessness

*Orbit:* If the launch speed of a projectile is sufficiently large, there comes a point at which the curve of the trajectory and the curve of the Earth are parallel. Such a *closed trajectory* is called an orbit.

An orbiting projectile is in free fall.

$v_"orbit" = sqrt(g r)$ - satellites need to maintain this speed to avoid falling into the planet.

== Newton's Law of Gravity

$ F_g = (G m_1 m_2)/r^2$

*Gravitational constant:* $G = 6.67 times 10^(-11) upright((N m^2) / "kg"^2)$

$ g_"planet" = (G M_"planet")/(R_"planet"^2)$

== Gravity and Orbits
