# 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) \(10 \pi \; rad \; s^{-1}\) |

**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!**

**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.

**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 \(1/T\)

**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!**