Simple Harmonic Motion (SHM) Practice Flashcards

A set of flashcards covering Simple Harmonic Motion (SHM) characteristics, equations, pendulum and mass-spring systems, damping, and resonance.

What are the two specific conditions for an object to be performing Simple Harmonic Motion (SHM)?

1. The acceleration of the object is always directed towards the equilibrium position. 2. The acceleration is always proportional to the displacement of the object from the equilibrium position ($a \propto -x$).

What is the defining equation for acceleration ($a$) in Simple Harmonic Motion?

$$a = -\omega^2x$$

How is the displacement ($x$) of an oscillator calculated in terms of amplitude ($A$) and time ($t$)?

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

What is the formula to calculate the velocity ($v$) of an object in SHM at any displacement ($x$)?

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

What are the expressions for maximum speed ($v_{max}$) and maximum acceleration ($a_{max}$)?

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

What are the mathematical relationships between the displacement-time ($x-t$), velocity-time ($v-t$), and acceleration-time ($a-t$) graphs?

The $v-t$ graph is derived from the gradient of the $x-t$ graph, and the $a-t$ graph is derived from the gradient of the $v-t$ graph.

What is the formula for the time period ($T$) of a mass-spring system?

$T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant.

What is the formula for the time period ($T$) of a simple pendulum?

$T = 2\pi \sqrt{\frac{l}{g}}$, where $l$ is the length and $g$ is the gravitational acceleration.

Under what condition is the simple pendulum equation for the time period valid?

It only applies at small amplitudes or angles where the small angle approximation ($\sin(x) \approx x$) is valid, typically less than $10^{\circ}$ or $12^{\circ}$.

What is the formula for the total energy ($E_{total}$) of a system in SHM?

$$E_{total} = \frac{1}{2}m\omega^2 A^2$$

How does the time period of an SHM oscillator change if the amplitude is increased?

The time period is independent of amplitude; it does not change if the maximum displacement is adjusted.

Define 'Free Vibration'.

A free vibration occurs when an object is displaced and then released to oscillate at its natural frequency ($f_0$) with no external forces acting on it.

Define 'Forced Vibration'.

An oscillation in which a periodic driving force is applied to a system, causing it to oscillate at the frequency of the driver.

What is 'Resonance'?

Resonance occurs when the driving frequency equals the natural frequency ($f_0$) of the system, resulting in oscillations with a very large amplitude.

What is the phase relationship between the driver and the driven system when the driving frequency ($f$) matches the natural frequency ($f_0$)?

The driven system is $90^{\circ}$ out of phase with the driver.

What are 'Damped Oscillations'?

Oscillations where a resistive force acts against the motion, causing energy to be dissipated and the amplitude to decrease over time.

What is 'Critical Damping'?

A type of damping where the system is returned to equilibrium in the shortest possible time without oscillating, commonly used in car suspensions.

During the pendulum experiment (Required Practical 7), where should the fiducial marker be placed and why?

The fiducial marker should be placed at the equilibrium position because this is where the pendulum moves fastest and spends the least time, allowing for more accurate timing.

In a power law experiment where $T = kl^n$, what does the gradient of a graph of $\ln(T)$ against $\ln(l)$ represent?

The gradient represents the power $n$ (which for a pendulum is $0.5$).

What four things happen to a resonance curve as the magnitude of damping increases?

1. The resonant frequency decreases. 2. The sharpness of the resonance peak decreases. 3. The amplitude at resonance decreases. 4. The amplitude is reduced at all driving frequencies.