# Amplitude, Period and Frequency

Here is a ball moving back and forth with simple harmonic motion (SHM):

Its position $x$ as a function of time $t$ is:

$x (t) = A \cdot \cos \Bigl( \frac {2\cdot \pi \cdot t}{T} \Bigr)$

where $A$ is the amplitude of motion: the distance from the centre of motion to either extreme

$T$ is the period of motion: the time for one complete cycle of the motion.

## Questions

Which ball has a larger amplitude?

Ball A or Ball B

Ball A - Correct!
Ball B - No. That is not correct

Which ball has a longer period?

Ball A or Ball B

Ball A - Correct!
Ball B - No. That is not correct

What is the period of Ball B?

A) 4.0s
B) 8.0s
C) 12 s
D) 16s

A) 4.0s - No. The period is the time for one full oscillation.
B) 8.0s - No. The period is the time for one full oscillation.
C) 12 s - Correct!
D) 16s - No. The period is the time for one full oscillation.

The frequency of motion, $f$, is the rate of repetition of the motion -- the number of cycles per unit time. There is a simple relation between frequency and period: $f = T^{-1}$

What is the frequency of ball B (recall, the period is 12s)?

A) 0.0625 Hz
B) 0.0833 Hz
C) 0.125 Hz
D) 0.250 Hz

A) 0.0625 Hz - No. Remember $f = 1/T$
B) 0.0833 Hz - Correct!
C) 0.125 Hz  - No. Remember $f = 1/T$
D) 0.250 Hz  - No. Remember $f = 1/T$

Angular frequency is the rotational analogy to frequency. Represented as , and is the rate of change of an angle when something is moving in a circular orbit. This is the usual frequency (measured in cycles per second), converted to radians per second. That is $\omega = 2\pi / T = 2\pi f$

Which ball has the larger angular frequency?

Ball A or Ball B

Ball A - No. $\omega$ is proportional to $f$

Ball B - Correct!

What is ball B's angular frequency?

A) $0.125\pi \; \mathrm {rad} \; s^{-1}$
B) $0.167\pi \; \mathrm {rad} \; s^{-1}$
C) $0.250\pi \; \mathrm {rad} \; s^{-1}$
D) $0.500\pi \; \mathrm {rad} \; s^{-1}$

A) $0.125\pi \; \mathrm {rad} \; s^{-1}$ - No. $\omega = 2pf$
B) $0.167\pi \; \mathrm {rad} \; s^{-1}$ - Correct!
C) $0.250\pi \; \mathrm {rad} \; s^{-1}$ -  No. $\omega = 2pf$
D) $0.500\pi \; \mathrm {rad} \; s^{-1}$ -  No. $\omega = 2pf$

From this graph, find the following:

1. Amplitude ($A$) 2. period ($T$) 3. frequency ($f$) 4. angular frequency ($\omega$)
A) 20 cm A) 0.20 s A) 0.20 Hz A) $0.20 \pi \; rad \; s^{-1}$
B) 1.0 cm B) 1.0 s B) 1.0 Hz B) $1.0 \pi \; rad \; s^{-1}$
C) 5.0 cm C) 5.0 s C) 5.0 Hz C) $5.0 \pi \; rad \; s^{-1}$
D) 10 cm D) 10 s D) 10 Hz D) $10 \pi \; rad \; s^{-1}$

1. amplitude ($A$):
A) 20 cm - No. It is not the distance from a crest to a trough.
B) 1.0 cm - No. That is not correct.
C) 5.0 cm - No. That is not correct.
D) 10 cm - Correct!

2. period ($T$):
A) 0.20 s - Correct!
B) 1.0 s - No. The period is the time for one full oscillation.
C) 5.0 s - No. The period is the time for one full oscillation.
D) 10 s - No. The period is the time for one full oscillation.

3. frequency ($f$):
A) 0.20 Hz - No. The frequency is $1/T$
B) 1.0 Hz - No. The frequency is $1/T$
C) 5.0 Hz - Correct!
D) 10 Hz - No. The frequency is $1/T$

4. angular frequency ($\omega$):
A) $0.20 \pi \; rad \; s^{-1}$ - No. $\omega = 2\pi f$
B) $1.0 \pi \; rad \; s^{-1}$ - No. $\omega = 2\pi f$
C) $5.0 \pi \; rad \; s^{-1}$ - No. $\omega = 2\pi f$
D) $10 \pi \; rad \; s^{-1}$ - Correct!