Amplitude, Period and Frequency

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

ball moving back and forth with simple harmonic motion

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. \(\omega = 2pf\)

 

Position-time graph of a moving piston
Figure 1: position-time graph of a piston moving in and out.

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!