Simple Harmonic Motion (SHM) Flashcards

A comprehensive set of practice questions covering the definitions, mathematical equations, systems, and phenomena of Simple Harmonic Motion as outlined in the lecture notes.

What are the two primary conditions for an object to be in 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.

What is the defining equation for acceleration in SHM?

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

How is the displacement ($x$) of an object in SHM calculated as a function of time ($t$)?

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

What is the relationship between velocity ($v$), angular frequency ($\omega$), amplitude ($A$), and displacement ($x$)?

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

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

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

What is the equation 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 equation 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.

According to the notes, what feature of simple harmonic motion is independent of the amplitude?

The time period ($T$) is independent of the amplitude.

Where should a fiducial marker be placed during a pendulum experiment and why?

It is placed in the equilibrium position because this is where the pendulum moves fastest and spends the least amount of time, allowing for a more accurate time establishing.

How is the total energy or maximum kinetic energy ($E_{kmax}$) of an SHM system calculated?

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

Define 'Free Vibrations'.

A free vibration is one in which there are no external forces; the object oscillates at its natural frequency ($f_0$) with constant amplitude and total energy.

Define 'Forced Oscillations'.

A forced oscillation is one in which a periodic driving force is applied to an oscillating system, causing the object to oscillate at the frequency of the driver.

What is resonance?

Resonance occurs when the driving frequency equals the natural frequency of the system being driven ($f = f_0$), resulting in the system oscillating with a very large amplitude.

What is the phase relationship between the driver and the driven system at resonance?

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

Describe the phase relationship when the driving frequency is much higher than the natural frequency ($f \gg f_0$).

The oscillations are $180^\circ$ out of phase.

Define light, heavy, and critical damping.

Light damping: Amplitude reduces gradually and the period is unchanged. Heavy damping: The period is longer and amplitude is reduced. Critical damping: The system returns to equilibrium in the shortest time possible without oscillating.

What are four effects of increasing damping on a resonance curve?

1. Resonant frequency decreases. 2. Sharpness of resonance decreases. 3. Amplitude of resonance decreases. 4. Amplitude is reduced at all frequencies.

How do you determine the power $n$ for the relationship $T \propto l^n$ using a log graph?

By plotting $\ln(T)$ against $\ln(l)$, where the gradient of the straight line equals $n$. For a pendulum, $n \approx 0.5$.