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 \(A = 10\)

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