# Simple Harmonic Motion Self Test

## Question 1

A large mass bobs up and down on a heavy spring. Initially the mass is at the top \(20 \;cm\) above its lowest position. The magnitude of its maximum speed is \(31 \;cm/s.\)

### What is the amplitude for this motion?

A. \(10 \;cm\)

B. \(20\; cm\)

C. \(31 \;cm\)

D. \(40 \;cm\)

E. \(62\; cm \)

**A. \(10 \;cm\) - Correct!**

B. \(20 \;cm\) - No. That is the total displacement.

C. \(31\; cm\) - No.

D. \(40 \;cm\) - No.

E. \(62 \;cm\) - No.

### What is the angular frequency for this motion?

A. \(0.00 \pi \; \mathrm {rad} \;s^{-1}\)

B. \(0.25 \pi \; \mathrm {rad} \;s^{-1}\)

C. \(0.50 \pi \; \mathrm {rad} \;s^{-1}\)

D. \(0.75 \pi \; \mathrm {rad} \;s^{-1}\)

E. \(1.0 \pi \; \mathrm {rad} \;s^{-1}\)

A. \(0.00 \pi \; \mathrm {rad} \;s^{-1}\) - No. If the speed is given by \(v(t) = -\omega A \sin(\omega t)\) what is its maximum value?

B. \(0.25 \pi \; \mathrm {rad} \;s^{-1}\) - No. If the speed is given by \(v(t) = -\omega A \sin(\omega t)\) what is its maximum value?

C. \(0.50 \pi \; \mathrm {rad} \;s^{-1}\) - No. If the speed is given by \(v(t) = -\omega A \sin(\omega t)\) what is its maximum value?

D. \(0.75 \pi \; \mathrm {rad} \;s^{-1}\) - No. If the speed is given by \(v(t) = -\omega A \sin(\omega t)\) what is its maximum value?

**E. \(1.0 \pi \; \mathrm {rad} \;s^{-1}\) - Correct!**

**\(v = -\omega A \sin \omega t\)**

**Maximum value when \(\sin = 1\)**

**\(|v_{max}| = \omega A\)**

**\(\omega = |v_{max}|/A = 31/10 = 3.1 = 1.0 \pi \; s^{-1} \)**

## Quesstion 2

A large mass bobs up and down on a heavy spring. Initially it is at the top. It achieves its maximum downward velocity of \(94 \;cm \; s^{-1}\) in \(0.25 s\) from its release.

### What are the period, angular freuqnecy, and amplitude for this motion?

\(T =\)

A. \(0.25 s\)

B. \(0.5 s\)

C. \(0.75 s\)

D. \(1.0 s\)

E. \(2.0 s\)

A. \(0.25 s\) - No. It hasnt gone through a full cycle.

B. \(0.5 s\) - No. It hasnt gone through a full cycle.

C. \(0.75 s\) - No. It hasnt gone through a full cycle.

**D. \(1.0 s\) - Correct!**

E. \(2.0 s\) - No. It has gone more than one full oscillation.

\(\omega =\)

A. \(0.00 \pi \; rad \; s^{-1}\)

B. \(0.50 \pi \; rad \; s^{-1}\)

C. \(1.0 \pi \; rad \; s^{-1}\)

D. \(1.5 \pi \; rad \; s^{-1}\)

**E. \(2.0 \pi \; rad \; s^{-1}\) **

A. \(0.00 \pi \; rad \; s^{-1}\) - No that is not correct. What is the relationship between \(\omega\) and \(T\)?

B. \(0.50 \pi \; rad \; s^{-1}\) - No that is not correct. What is the relationship between \(\omega\) and \(T\)?

C. \(1.0 \pi \; rad \; s^{-1}\) - No that is not correct. What is the relationship between \(\omega\) and \(T\)?

D. \(1.5 \pi \; rad \; s^{-1}\) - No that is not correct. What is the relationship between \(\omega\) and \(T\)?

**E. \(2.0 \pi \; rad \; s^{-1}\) - Correct!**

**\(\omega = 2\pi /T = 2 \pi/1 = 2\pi \;s^{-1}\) **

\(A =\)

A. \(15.0\; cm\)

B. \(23.0 \;cm\)

C. \(94.0 \;cm\)

D. \(156 \;cm\)

E. \(188 \;cm\)

**A. \(15.0\; cm\) - Correct!**

B. \(23.0 \;cm\) - No. What is the expression for the maximum velocity?

C. \(94.0 \;cm\) - No. What is the expression for the maximum velocity?

D. \(156 \;cm\) - No. What is the expression for the maximum velocity?

E. \(188 \;cm\) - No. What is the expression for the maximum velocity?

\(v_{max} = -\omega A\)

\(A = v_{max}/\omega = 94/2\pi = 15 \; \mathrm cm\)

## Question 3

A large mass bobs up and down on a heavy spring. At first, it is seen moving up with a maximum velocity of \(47 \;cm/s.\) When it is \(8.7 \;cm\) above the central position, its speed is half its maximum value.

### What are the amplitude, and angular frequency for this motion?

\(A =\)

A. \(5.0 \;cm\)

B. \(10 \;cm\)

C. \(15 \;cm\)

D. \(20\; cm\)

E. \(25 \;cm\)

A. \(5.0 \;cm\) - No. That is not correct. What are the expressions for the velocity and the maximum velocity?

**B. \(10\; cm\) - Correct!**

C. \(15 \;cm\) - No. That is not correct. What are the expressions for the velocity and the maximum velocity?

D. \(20 \;cm\) - No. That is not correct. What are the expressions for the velocity and the maximum velocity?

E. \(25 \;cm\) - No. That is not correct. What are the expressions for the velocity and the maximum velocity?

\(v = -\omega A \sin \omega t\)

\(v_{max} = -\omega A = 47\)

\(v = 47 \sin \omega t\)

\(23.5 = 47 \sin \omega t\)

\(\sin \omega t = 1/2\)

\(\omega t = 30^ \circ\)

\(x = A \cos \omega t\)

\(8.7 = A \cos 30 = 0.866\)

\(A = 8.7/0.866 = 10 \mathrm {cm}\)

\(\omega =\)

A. \(0.00 \pi \; rad \; s^{-1}\)

B. \(0.50 \pi \; rad \; s^{-1}\)

C. \(1.0 \pi \; rad \; s^{-1}\)

D. \(1.5 \pi \; rad \; s^{-1}\)

**E. \(2.0 \pi \; rad \; s^{-1}\) **

A. \(0.00 \pi \; rad \; s^{-1}\) - No. What is the relation for \(v_{max}\)?

**B. \(0.50 \pi \; rad \; s^{-1}\) - Correct!**

C. \(1.0 \pi \; rad \; s^{-1}\) - No. What is the relation for \(v_{max}\)?

D. \(1.5 \pi \; rad \; s^{-1}\) - No. What is the relation for \(v_{max}\)?

**E. \(2.0 \pi \; rad \; s^{-1}\) - **No. What is the relation for \(v_{max}\)?

\(v_{max} = -\omega A\)

but \(A = 10\)

\(\omega = 47/10 = 0.75 (2\pi)\)