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