Oscillations: Physics 9702 (Cambridge International AS & A Level)

A small ball attached to a horizontal spring oscillates with simple harmonic motion of amplitude $x_0 = 4.5\text{ cm}$ and frequency $f = 3.0\text{ Hz}$.

What is the maximum speed of the ball during the oscillation?

A mass of $m = 0.250\text{ kg}$ is attached to the end of a spring of force constant $k = 40\text{ N m}^{-1}$ and hangs in equilibrium. The mass is then pulled down a further $6.0\text{ cm}$ from its equilibrium position and released from rest at time $t=0$, so that it oscillates with simple harmonic motion of amplitude $x_0 = 0.060\text{ m}$.

(a) State the equation for the displacement $x$ of the mass at time $t$ after release, and describe the shape of the graph of $x$ against $t$ that this equation represents over one complete oscillation. [2]

(b) Show that the angular frequency $\omega$ of the oscillation is $12.6\text{ rad s}^{-1}$. [2]

(c) Calculate the maximum speed of the mass during the oscillation. [2]

(d) Calculate the maximum magnitude of the acceleration of the mass during the oscillation. [2]

A simple pendulum consists of a small bob of mass $m = 0.120\text{ kg}$ on a light string of length $L = 0.900\text{ m}$. The bob is displaced sideways to give an amplitude of $x_0 = 5.0\text{ cm}$ and released from rest, so that it swings with simple harmonic motion. Take $g = 9.81\text{ m s}^{-2}$.

(a) Show that the period of oscillation is $1.90\text{ s}$. [2]

(b) Calculate the angular frequency $\omega$ of the oscillation. [2]

(c) Calculate the total energy of the oscillation. [2]

(d) Calculate the kinetic energy and the potential energy of the bob at the instant its displacement from the equilibrium position is $3.0\text{ cm}$. [3]

An engineering student sets up a demonstration rig: a metal block hangs from a vertical spring, with the block partly submerged in a cylinder of oil so that its oscillations can be damped by varying the oil's viscosity. In each case below, the block is pulled down by the same amount and released from rest.

(a) With no oil in the cylinder, the block oscillates with negligible damping. State what is meant by damping in an oscillating system, and describe how adding light damping would change the displacement-time graph of the block compared with this undamped case. [3]

(b) Oil is added so that the block returns to its equilibrium position in the shortest possible time, without overshooting and oscillating about it. State the name given to this particular case of damping, and describe how heavy damping would differ from it. [3]

(c) A small motor is later attached to the top of the spring, shaking the support up and down at an adjustable frequency. Explain what is meant by resonance, and state the condition on the driving frequency required for resonance to occur in this system. [2]

A particle oscillates with simple harmonic motion of amplitude $x_0 = 0.12\text{ m}$ and period $T = 0.50\text{ s}$.

What is the speed of the particle at the instant its displacement from the centre of the oscillation is $x = 0.072\text{ m}$?

The cone of a small loudspeaker oscillates with simple harmonic motion of amplitude $x_0 = 1.5\text{ mm}$ and frequency $f = 200\text{ Hz}$.

What is the magnitude of the maximum acceleration of the cone during the oscillation?

A prong of a vibrating tuning fork oscillates with simple harmonic motion of amplitude $x_0 = 0.80\text{ mm}$ and frequency $f = 256\text{ Hz}$. At time $t=0$, the tip of the prong passes through the centre of its oscillation moving with its maximum speed.

(a) State the appropriate equation for the displacement $x$ of the prong tip at time $t$, and explain why this form (rather than a cosine equation) applies here. [2]

(b) Calculate the angular frequency $\omega$ of the oscillation. [2]

(c) Calculate the displacement and the velocity of the prong tip at $t = 2.0\times10^{-4}\text{ s}$. [3]

(d) Use $a=-\omega^2x$ to calculate the magnitude of the acceleration of the prong tip at this same instant, and state the direction of this acceleration relative to the displacement found in (c). [2]

A block of mass $m = 0.200\text{ kg}$ rests on a frictionless horizontal surface and is attached to a light spring of force constant $k = 18\text{ N m}^{-1}$. The block is pulled aside by $x_0 = 5.0\text{ cm}$ from its equilibrium position and released from rest, so that it oscillates with simple harmonic motion of amplitude $x_0 = 0.050\text{ m}$.

(a) Show that the angular frequency of the oscillation is $\omega = 9.49\text{ rad s}^{-1}$. [2]

(b) Calculate the total energy of the oscillation. [2]

(c) Show that the displacement at which the kinetic energy of the block is exactly twice its potential energy is $x = 2.89\times10^{-2}\text{ m}$. [3]

(d) Calculate the speed of the block at this displacement. [2]

A displacement-time graph for a lightly damped oscillating system shows the amplitude of successive oscillations gradually decreasing over time.

Which one of the following statements about this system is correct?

A particle P moves with simple harmonic motion such that its displacement is $x=x_0\cos\omega t$, where $x_0 = 5.0\text{ cm}$ and $\omega = 4.0\text{ rad s}^{-1}$.

(a) Calculate the period $T$ of the motion, and state the values of the displacement $x$, velocity $v$ and acceleration $a$ of P at $t=0$. [2]

(b) Calculate the values of $x$, $v$ and $a$ of P at $t=T/4$. [2]

(c) Calculate the values of $x$, $v$ and $a$ of P at $t=T/2$. [2]

(d) Using your answers to (a)-(c), describe the phase relationship of the velocity-time graph relative to the displacement-time graph, and of the acceleration-time graph relative to the displacement-time graph. [3]