SHM flash cards

1. 1.What is periodic motion?

   Motion that repeats itself at regular intervals of time (e.g. motion of hands of a clock, planets around the Sun).

   Hint: Repeats after a fixed period.

2. 2.What is oscillatory (vibratory) motion?

   To-and-fro motion of a body about a fixed mean (equilibrium) position. Every oscillatory motion is periodic, but not every periodic motion is oscillatory.

   Hint: To-and-fro about equilibrium.

3. 3.Define Simple Harmonic Motion (SHM).

   Oscillatory motion in which the restoring force (or acceleration) is directly proportional to the displacement from the mean position and is always directed towards it: $F = -kx$.

   Hint: $F \propto -x$.

4. 4.Write the defining differential equation of SHM.

   $\dfrac{d^2x}{dt^2} + \omega^2 x = 0$, where $\omega^2 = k/m$ is the square of the angular frequency.

   Hint: Second-order, restoring term $\omega^2 x$.

5. 5.What is the general displacement equation of a particle in SHM?

   $x = A\sin(\omega t + \phi)$ or equivalently $x = A\cos(\omega t + \phi)$, where $A$ is amplitude, $\omega$ angular frequency and $\phi$ initial phase.

   Hint: Sine/cosine of $(\omega t + \phi)$.

6. 6.What does the amplitude $A$ represent in SHM?

   The maximum displacement of the particle from its mean position. It fixes the energy of the oscillation but not the period.

   Hint: Peak displacement.

7. 7.Define the time period $T$ of SHM and relate it to $\omega$.

   $T$ is the time for one complete oscillation. $T = \dfrac{2\pi}{\omega}$.

   Hint: $2\pi/\omega$.

8. 8.Define frequency $f$ and relate it to $\omega$ and $T$.

   $f$ is the number of oscillations per second. $f = \dfrac{1}{T} = \dfrac{\omega}{2\pi}$, unit hertz (Hz).

   Hint: $1/T = \omega/2\pi$.

9. 9.What is angular frequency $\omega$?

   $\omega = 2\pi f = \dfrac{2\pi}{T}$; it measures the rate of phase change, unit rad s$^{-1}$.

   Hint: $2\pi f$.

10. 10.What is the phase $(\omega t + \phi)$ of SHM?

    The quantity that determines the state (position and direction of motion) of the oscillator at a given instant. $\phi$ is the phase constant (initial phase / epoch).

    Hint: State of oscillator at time $t$.

11. 11.Write the velocity of a particle in SHM as a function of time (for $x=A\sin\omega t$).

    $v = \dfrac{dx}{dt} = A\omega\cos\omega t$; maximum speed $v_{max} = A\omega$ at the mean position.

    Hint: Differentiate $x$.

12. 12.Express velocity in SHM in terms of displacement $x$.

    $v = \omega\sqrt{A^2 - x^2}$. It is maximum ($A\omega$) at $x=0$ and zero at $x=\pm A$.

    Hint: $\omega\sqrt{A^2-x^2}$.

13. 13.Write the acceleration of a particle in SHM (for $x=A\sin\omega t$).

    $a = \dfrac{d^2x}{dt^2} = -A\omega^2\sin\omega t = -\omega^2 x$; maximum magnitude $a_{max} = A\omega^2$ at the extremes.

    Hint: $-\omega^2 x$.

14. 14.Where in SHM are velocity and acceleration each maximum/minimum?

    Velocity is maximum at the mean position and zero at extremes; acceleration is zero at the mean position and maximum at the extremes.

    Hint: They swap roles at mean vs extreme.

15. 15.What is the phase difference between displacement and velocity in SHM?

    Velocity leads displacement by $\dfrac{\pi}{2}$ (90°).

    Hint: $\pi/2$ ahead.

16. 16.What is the phase difference between displacement and acceleration in SHM?

    Acceleration is out of phase with displacement by $\pi$ (180°); $a=-\omega^2 x$.

    Hint: Opposite direction, $\pi$.

17. 17.Show the shape of the velocity–displacement curve in SHM.

    From $v=\omega\sqrt{A^2-x^2}$, we get $\dfrac{v^2}{A^2\omega^2}+\dfrac{x^2}{A^2}=1$: an ellipse.

    Hint: Ellipse.

18. 18.Show the shape of the acceleration–displacement curve in SHM.

    Since $a=-\omega^2 x$, it is a straight line through the origin with negative slope $-\omega^2$.

    Hint: Straight line, slope $-\omega^2$.

19. 19.How is SHM related to uniform circular motion?

    SHM is the projection of uniform circular motion on a diameter of the reference circle. The radius equals the amplitude and the angular speed equals $\omega$.

    Hint: Projection of the reference circle.

20. 20.What is the restoring force in SHM?

    The force that always acts towards the mean position and is proportional to displacement: $F=-kx$; $k$ is the force constant.

    Hint: $-kx$, towards mean.

21. 21.Relate $\omega$, $k$ and $m$ for a linear SHM oscillator.

    $\omega = \sqrt{\dfrac{k}{m}}$, so $T = 2\pi\sqrt{\dfrac{m}{k}}$.

    Hint: $\sqrt{k/m}$.

22. 22.Write the kinetic energy of a particle in SHM as a function of $x$.

    $KE = \dfrac{1}{2}m\omega^2(A^2 - x^2)$; maximum at the mean position.

    Hint: $\tfrac12 m\omega^2(A^2-x^2)$.

23. 23.Write the potential energy of a particle in SHM as a function of $x$.

    $PE = \dfrac{1}{2}m\omega^2 x^2 = \dfrac{1}{2}kx^2$; maximum at the extremes.

    Hint: $\tfrac12 kx^2$.

24. 24.Write the total mechanical energy in SHM.

    $E = KE + PE = \dfrac{1}{2}m\omega^2 A^2 = \dfrac{1}{2}kA^2$; it is constant and independent of $x$.

    Hint: $\tfrac12 m\omega^2 A^2$, constant.

25. 25.How does the total energy of SHM depend on amplitude and frequency?

    $E \propto A^2$ and $E \propto \omega^2$ (i.e. $\propto f^2$). Doubling the amplitude quadruples the energy.

    Hint: $E\propto A^2\omega^2$.

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