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