# Equations for SHM

It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!

However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.

Key Concepts

By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.

The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.

Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.

Essentials

Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.

This can be used to show that $$a=-\omega^2x$$.

Summary

The equation for the displacement of a body executing SHM is either:

• $$x = x_o\sin(2πft)$$
• $$x = x_o\cos(2πft)$$

Test Yourself

Use quizzes to practise application of theory.

Exam-style Questions
The passage provides an explanation of simple harmonic motion (SHM), including the relationship between displacement, velocity, and acceleration, and how these quantities are represented mathematically. It also includes a test question prompt at the end.
no_question

the passage does not contain a specific mathematical question or a provided answer. the "test yourself" section suggests the presence of quizzes and exam-style questions, but no actual questions or answers are included in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Equations for SHM

It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!

However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.

Key Concepts

By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.

The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.

Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.

Essentials

Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.

This can be used to show that $$a=-\omega^2x$$.

Summary

The equation for the displacement of a body executing SHM is either:

• $$x = x_o\sin(2πft)$$
• $$x = x_o\cos(2πft)$$

Test Yourself

Use quizzes to practise application of theory.

Exam-style Questions
user
The passage provides an explanation of simple harmonic motion (SHM), including the relationship between displacement, velocity, and acceleration, and how these quantities are represented mathematically. It also includes a test question prompt at the end.

TYPE: NO_QUESTION

The passage does not contain a specific mathematical question or a provided answer. The "Test Yourself" section suggests the presence of quizzes and exam-style questions, but no actual questions or answers are included in the text.