Simple Harmonic Motion (SHM) and Waves — Hard Practice Quiz

A Physics cheat sheet for Simple Harmonic Motion (SHM) and Waves — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. If the period of a mass-spring system is halved while the oscillating mass remains constant, how must the amplitude change to keep the total mechanical energy of oscillation the same?

   • The amplitude must be halved.
   • The amplitude must be quartered.
   • The amplitude must be doubled.
   • The amplitude must be quadrupled.

   Answer: The amplitude must be halved.

2. An object undergoing Simple Harmonic Motion (SHM) has a maximum velocity of $v_{max}$ and a maximum acceleration of $a_{max}$. What is the amplitude of its oscillation?

   • $A = v_{max}^2 / a_{max}$
   • $A = a_{max}^2 / v_{max}$
   • $A = v_{max} / a_{max}$
   • $A = \sqrt{v_{max} / a_{max}}$

   Answer: $A = v_{max}^2 / a_{max}$

3. An object in SHM has an amplitude $A$. At a certain instant, its position is $x$ and its speed is $v$. Which expression correctly represents its angular frequency $\omega$?

   • $\omega = v / \sqrt{A^2 - x^2}$
   • $\omega = \sqrt{v^2 - x^2} / A$
   • $\omega = A / \sqrt{v^2 - x^2}$
   • $\omega = \sqrt{A^2 - x^2} / v$

   Answer: $\omega = v / \sqrt{A^2 - x^2}$

4. A simple pendulum has a period $T$ on Earth, where the acceleration due to gravity is $g$. If the pendulum is taken to a planet where the acceleration due to gravity is $g/4$ and its length is doubled, what will its new period be?

   • $T' = 2\sqrt{2} T$
   • $T' = \frac{T}{2\sqrt{2}}$
   • $T' = 4T$
   • $T' = \frac{T}{4}$

   Answer: $T' = 2\sqrt{2} T$

5. A wave travels at a speed $v$ and has a wavelength $\lambda$. If this wave is generated by an oscillator undergoing SHM, what is the maximum acceleration of a particle in the medium if the amplitude of oscillation is $A$?

   • $a_{max} = A (4\pi^2 v^2 / \lambda^2)$
   • $a_{max} = A (2\pi v / \lambda)$
   • $a_{max} = A (v^2 / \lambda^2)$
   • $a_{max} = A (v / \lambda)^2$

   Answer: $a_{max} = A (4\pi^2 v^2 / \lambda^2)$

6. A mass $m$ is suspended from a wire of length $L_0$, cross-sectional area $A_{cs}$, and Young's modulus $Y$. If the mass is displaced vertically and oscillates, what is the period of its oscillation?

   • $T_s = 2\pi \sqrt{mL_0 / (YA_{cs})}$
   • $T_s = 2\pi \sqrt{mY / (L_0 A_{cs})}$
   • $T_s = 2\pi \sqrt{mA_{cs} / (YL_0)}$
   • $T_s = 2\pi \sqrt{m / (YA_{cs}L_0)}$

   Answer: $T_s = 2\pi \sqrt{mL_0 / (YA_{cs})}$

7. For a mass-spring system undergoing SHM, express the total mechanical energy $E$ in terms of the oscillating mass $m$ and its maximum velocity $v_{max}$.

   • $E = \frac{1}{2}m v_{max}^2$
   • $E = m v_{max}^2$
   • $E = \frac{1}{2}m v_{max}^2 \omega^2$
   • $E = \frac{1}{2}m v_{max}^2 T^2$

   Answer: $E = \frac{1}{2}m v_{max}^2$

8. A fluid with initial density $\rho_0$ is subjected to a pressure increase $\Delta P$. If its Bulk Modulus is $B$, what is the new density $\rho$ in terms of $\rho_0$, $B$, and $\Delta P$?

   • $\rho = \rho_0 \frac{B}{B - \Delta P}$
   • $\rho = \rho_0 \frac{B + \Delta P}{B}$
   • $\rho = \rho_0 \frac{B - \Delta P}{B}$
   • $\rho = \rho_0 \frac{B}{B + \Delta P}$

   Answer: $\rho = \rho_0 \frac{B}{B - \Delta P}$

9. An object undergoes SHM. Its maximum acceleration is $a_{max}$ and its maximum velocity is $v_{max}$. What is its period of oscillation $T$?

   • $T = 2\pi v_{max} / a_{max}$
   • $T = 2\pi a_{max} / v_{max}$
   • $T = (v_{max} / a_{max}) / (2\pi)$
   • $T = 2\pi \sqrt{v_{max} / a_{max}}$

   Answer: $T = 2\pi v_{max} / a_{max}$

10. Consider a mass-spring system undergoing SHM. If the total mechanical energy of the system is doubled, and the oscillating mass is also doubled, how does the period of oscillation change if the amplitude remains constant?

    • The period remains unchanged.
    • The period increases by a factor of $\sqrt{2}$.
    • The period decreases by a factor of $\sqrt{2}$.
    • The period doubles.

    Answer: The period remains unchanged.