Simple Harmonic Motion (SHM) and Waves — Practice Quiz

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

Formulas & key concepts

Hooke's Law: Restoring force \(F\) of a spring is proportional to displacement \(x\).

$$F = -kx$$

Where: \(k\) = spring constant

Position in SHM: Displacement \(x\) as a function of time \(t\) (assuming phase \(\phi=0\)).

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

Velocity in SHM: Velocity \(v\) as a function of time.

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

Acceleration in SHM: Acceleration \(a\) as a function of time.

$$a(t) = -A\omega^2 \cos(\omega t)$$

Angular Frequency: Relates angular frequency \(\omega\) to frequency \(f\) and period \(T\).

$$\omega = 2\pi f = \frac{2\pi}{T}$$

Maximum Velocity: The maximum speed of the oscillating object.

$$v_{max} = A\omega$$

Maximum Acceleration: The maximum acceleration of the oscillating object.

$$a_{max} = A\omega^2$$

Period of Mass-Spring System: Period \(T_s\) depends on mass \(m\) and spring constant \(k\).

$$T_s = 2\pi \sqrt{\frac{m}{k}}$$

Period of Simple Pendulum: Period \(T_p\) depends on length \(L\) and gravity \(g\).

$$T_p = 2\pi \sqrt{\frac{L}{g}}$$

Energy in SHM: Total mechanical energy is proportional to the square of the amplitude.

$$E = \frac{1}{2}kA^2$$

Young's Modulus: Relates force \(F\) to stretch/compression \(\Delta L\).

$$F = Y(\frac{\Delta L}{L_0})A$$

Where: \(Y\) = Young's Modulus

Shear Modulus: Relates shear force \(F\) to shear deformation \(\Delta x\).

$$F = S(\frac{\Delta x}{L_0})A$$

Where: \(S\) = Shear Modulus

Bulk Modulus: Relates change in pressure \(\Delta P\) to fractional volume change.

$$\Delta P = -B(\frac{\Delta V}{V_0})$$

Where: \(B\) = Bulk Modulus

Wave Speed Equation: Frequency \(f\) equals speed \(v\) divided by wavelength \(\lambda\).

$$f = \frac{v}{\lambda}$$

Practice quiz

  1. A spring stretches by $0.1 \text{ m}$ when a $2 \text{ kg}$ mass is hung from it. Assuming $g = 9.8 \text{ m/s}^2$, what is the spring constant $k$?

    • $19.6 \text{ N/m}$
    • $196 \text{ N/m}$
    • $1.96 \text{ N/m}$
    • $0.196 \text{ N/m}$

    Answer: $196 \text{ N/m}$

  2. If the mass attached to a spring in a simple harmonic motion system is quadrupled, how does the period of oscillation $T_s$ change?

    • It doubles.
    • It halves.
    • It quadruples.
    • It remains the same.

    Answer: It doubles.

  3. A simple pendulum has a period of $2 \text{ s}$ on Earth. If it is moved to a planet where the acceleration due to gravity is $4$ times that on Earth, what will its new period be?

    • $0.5 \text{ s}$
    • $1 \text{ s}$
    • $2 \text{ s}$
    • $4 \text{ s}$

    Answer: $1 \text{ s}$

  4. An object undergoing simple harmonic motion has an amplitude $A = 0.05 \text{ m}$ and an angular frequency $\omega = 10 \text{ rad/s}$. What is its maximum speed $v_{max}$?

    • $0.005 \text{ m/s}$
    • $0.05 \text{ m/s}$
    • $0.5 \text{ m/s}$
    • $5 \text{ m/s}$

    Answer: $0.5 \text{ m/s}$

  5. An object in SHM has an amplitude $A = 0.02 \text{ m}$ and a maximum velocity $v_{max} = 0.4 \text{ m/s}$. What is its maximum acceleration $a_{max}$?

    • $8 \text{ m/s}^2$
    • $0.8 \text{ m/s}^2$
    • $0.08 \text{ m/s}^2$
    • $80 \text{ m/s}^2$

    Answer: $8 \text{ m/s}^2$

  6. A spring with a spring constant $k = 200 \text{ N/m}$ oscillates with an amplitude $A = 0.1 \text{ m}$. What is the total mechanical energy $E$ of the system?

    • $1 \text{ J}$
    • $2 \text{ J}$
    • $0.5 \text{ J}$
    • $10 \text{ J}$

    Answer: $1 \text{ J}$

  7. If the frequency $f$ of an oscillation is $5 \text{ Hz}$, what is its angular frequency $\omega$?

    • $5\pi \text{ rad/s}$
    • $10\pi \text{ rad/s}$
    • $2.5\pi \text{ rad/s}$
    • $20\pi \text{ rad/s}$

    Answer: $10\pi \text{ rad/s}$

  8. A wave has a frequency $f = 20 \text{ Hz}$ and a wavelength $\lambda = 0.5 \text{ m}$. What is its speed $v$?

    • $10 \text{ m/s}$
    • $40 \text{ m/s}$
    • $0.025 \text{ m/s}$
    • $100 \text{ m/s}$

    Answer: $10 \text{ m/s}$

  9. Young's Modulus $Y$ is a measure of a material's resistance to which type of deformation?

    • Shear deformation
    • Volume compression
    • Tensile or compressive stress (stretching or compression)
    • Torsional twisting

    Answer: Tensile or compressive stress (stretching or compression)

  10. At what point in its oscillation is the displacement $x(t)$ of an object in simple harmonic motion equal to its amplitude $A$?

    • When its velocity is maximum.
    • When its acceleration is maximum.
    • When its kinetic energy is maximum.
    • When it passes through the equilibrium position.

    Answer: When its acceleration is maximum.

Select a subject

Select a subject from the left panel to begin exploring formulas.