Damped and forced oscillations, resonance

One end of a uniform beam is clamped in position. A metal block of mass $m$ is attached to the free end, so the beam bends, as illustrated in Fig. 3.1. The block is then given a small vertical displacement and released, after which it oscillates with simple harmonic motion. The acceleration $a$ of the block is described by $a = -\frac{k}{m}x$, where $k$ is a constant for the beam and $x$ is the vertical displacement of the block from its equilibrium position.

Feb/March 2017

Explain the meaning of the natural frequency of vibration of a system.

Feb/March 2018

A body is moving in simple harmonic motion. Figure 3.1 shows how its velocity $v$ varies with displacement $x$.

Feb/March 2020

A long length of springy steel is held firmly at one end, leaving the strip upright. A mass of $65\,\text{g}$ is attached to the free end of the strip, as shown in Fig. 2.1. The mass is displaced to one side and then let go. Fig. 2.2 shows how the horizontal displacement of the mass changes with time $t$. The mass undergoes damped simple harmonic motion.

May/June 2010

A strip of springy steel is fixed at one end so that it stands vertically. A mass of $65\,\text{g}$ is fastened to the loose end of the strip, as shown in Fig. 2.1. The mass is pulled sideways and then let go. Fig. 2.2 shows how the horizontal displacement of the mass changes with time $t$. The mass carries out damped simple harmonic motion.

May/June 2010

Explain briefly the key principles behind using magnetic resonance to obtain diagnostic information about internal body structures.

May/June 2011

For an oscillating body, state what is meant by

May/June 2015

For an oscillating body, state the meaning of

May/June 2015

State, with reference to displacement, the meaning of simple harmonic motion.

May/June 2016

A metal block is suspended vertically from one end of a spring, while the spring’s other end is fastened to a thread that runs over a pulley and is connected to a vibrator, as illustrated in Fig. 4.1.

May/June 2016

State, with reference to displacement, what simple harmonic motion means.

May/June 2016

A bar magnet with mass $180\ \text{g}$ is hung from the free end of a spring, as shown in Fig. 2.1. It is positioned so that one pole lies close to the centre of a wire coil. The coil is connected in series with a resistor and a switch, and the switch is open. The magnet is moved vertically and then released so that it oscillates, with one pole still inside the coil while the other remains outside it. At time $t = 0$, the magnet is moving freely as it passes through its equilibrium position. At time $t = 3.0\ \text{s}$, the switch in the circuit is closed. Fig. 2.2 shows how the vertical displacement $y$ of the magnet varies with time $t$.

May/June 2017

A bar magnet with mass $250\,\text{g}$ is hung from the free end of a spring, as shown in Fig. 3.1. One pole of the magnet lies close to the middle of a coil of wire. The coil is linked in series with a resistor and a switch. The switch is open. The magnet is moved vertically and then left to oscillate, with one pole kept inside the coil. The other pole stays outside the coil. At time $t = 0$, the magnet is oscillating freely as it moves through its equilibrium position. At time $t = 6.0\,\text{s}$, the switch in the circuit is closed.

May/June 2017

A bar magnet with mass $180\,\text{g}$ is hung from the free end of a spring, as shown in Fig. 2.1. The magnet is positioned so that one pole lies close to the middle of a coil of wire. The coil is joined in series with a resistor and a switch, and the switch is open. The magnet is moved vertically and then released so that it oscillates, with one pole staying inside the coil and the other pole remaining outside it. At time $t = 0$, the magnet is oscillating freely as it goes through its equilibrium position. At time $t = 3.0\,\text{s}$, the switch in the circuit is closed. The graph in Fig. 2.2 shows how the vertical displacement $y$ of the magnet varies with time $t$.

May/June 2017

Describe the main principles involved in using nuclear magnetic resonance imaging (NMRI) to obtain information about structures inside the body.

May/June 2017

State two conditions that a mass must satisfy to be in simple harmonic motion.

May/June 2018

State the meaning of simple harmonic motion.

May/June 2021

A liquid occupies a U-shaped tube. In each limb of the tube, the liquid column has length $L$, as shown in Fig. 3.1. The liquid columns are shifted vertically. The liquid then oscillates in the tube. The liquid levels are shown displaced from their equilibrium positions in Fig. 3.2. The acceleration $a$ of the liquid in the tube is linked to the displacement $x$ by $a = -\left(\frac{g}{L}\right)x$, where $g$ is the acceleration of free fall.

May/June 2021

State the meaning of simple harmonic motion.

May/June 2021

State the meaning of resonance.

May/June 2022

State the meaning of resonance.

May/June 2024

State what resonance means.

May/June 2024

A bar magnet hangs from a spring. As shown in Fig. 7.1, one pole of the magnet swings freely inside a coil of wire. The switch $S$ is open at the start.

May/June 2025

A student arranges the apparatus shown in Fig. 3.1 to study the oscillations of a metal cube hanging from a spring. The oscillator produces vibrations with constant amplitude. Fig. 3.2 shows how the amplitude of the metal cube’s oscillations changes with frequency.

Oct/Nov 2010

A student assembles the apparatus shown in Fig. 3.1 to study the oscillations of a metal cube hung from a spring. The oscillator produces vibrations with a constant amplitude. Fig. 3.2 shows how the amplitude of the metal cube’s oscillations changes with frequency.

Oct/Nov 2010

A bar magnet hangs from the free end of a helical spring, as shown in Fig. 3.1. One pole of the magnet is inside a coil of wire. The coil is connected in series with a switch and a resistor. The switch is open. The magnet is moved vertically and then released. As the magnet passes through its equilibrium position, a timer is started. Fig. 3.2 shows how the vertical displacement y of the magnet from its equilibrium position varies with time t. At time t = 4.0\,\text{s}, the switch is closed.

Oct/Nov 2011

A bar magnet is hung from the free end of a helical spring, as shown in Fig. 3.1. One pole of the magnet is placed inside a coil of wire. The coil is connected in series with a switch and a resistor. The switch is open. The magnet is moved vertically and then let go. As the magnet passes through its rest position, a timer is started. Fig. 3.2 shows how the vertical displacement y of the magnet from its rest position varies with time t. At time t = 4.0\,\text{s}, the switch is closed.

Oct/Nov 2011

Fig. 13.1 shows a simplified block diagram of a mobile phone handset.

Oct/Nov 2014

Fig. 13.1 shows a simplified block diagram of a mobile phone handset.

Oct/Nov 2014

A light spring hangs from a fixed point, and a bar magnet is fixed to the lower end of the spring, as shown in Fig. 1.1. To protect the magnet from draughts, a cardboard cup is positioned around it without making contact. The magnet is then displaced vertically and released. Fig. 1.2 shows the way the vertical displacement $y$ of the magnet varies with time $t$.

Oct/Nov 2014

Distinguish free oscillations from forced oscillations.

Oct/Nov 2015

Distinguish between free oscillations and forced oscillations.

Oct/Nov 2015

As illustrated in Fig. 3.1, a U-tube holds liquid. The overall length of the liquid column in the tube is $L$. The column is then shifted so that the liquid level in each limb of the U-tube changes by $x$, as shown in Fig. 3.2. The liquid in the U-tube then carries out simple harmonic motion with acceleration $a$ given by $a = -\left(\frac{2g}{L}\right)x$, where $g$ is the acceleration of free fall.

Oct/Nov 2018

A U-tube is filled with liquid, as illustrated in Fig. 4.1. The overall length of the liquid column is $L$. The column is then moved so that the change in level from the equilibrium position in each arm of the U-tube is $x$, as shown in Fig. 4.2. The liquid in the U-tube then oscillates with acceleration $a$ given by $a = -\left( \frac{2g}{L} \right) x$, where $g$ is the acceleration of free fall.

Oct/Nov 2018

A U-tube is shown in Fig. 3.1 containing a liquid. The full length of the liquid column in the tube is $L$. The column is then shifted so that the rise or fall of the liquid in each arm of the U-tube is $x$, as shown in Fig. 3.2. The liquid in the U-tube then carries out simple harmonic motion, with the column acceleration $a$ given by $a = -\left(\frac{2g}{L}\right)x$, where $g$ is the acceleration of free fall.

Oct/Nov 2018

A ball of mass $M$ is supported on a horizontal surface by two identical extended springs, as shown in Fig. 4.1. One spring is connected to a fixed point. The other spring is connected to an oscillator. The oscillator is switched off. The ball is displaced sideways along the axis of the springs and is then released. Figure 4.2 shows how the displacement $x$ of the ball varies with time $t$.

Oct/Nov 2019

A trolley moving along a track is linked by springs to fixed blocks $X$ and $Y$, as shown in Fig. 4.1. The track has many tiny holes through which air is blown vertically upwards. This means the trolley rests on a cushion of air instead of touching the track directly. The trolley is drawn to one side of its equilibrium position and then released, so that at first it oscillates with simple harmonic motion. After a short time, the air blower is turned off. The way the distance $L$ of the trolley from block $X$ changes with time $t$ is shown in Fig. 4.2.

Oct/Nov 2021

A trolley moving along a track is connected by springs to fixed blocks $X$ and $Y$, as shown in Fig. 4.1. The track has many tiny holes through which air is driven vertically upward. This makes the trolley ride on a cushion of air instead of touching the track directly. The trolley is displaced to one side of its equilibrium position and then released, so that it first oscillates with simple harmonic motion. After a short time, the air blower is turned off. Fig. 4.2 shows how the distance $L$ of the trolley from block $X$ varies with time $t$.

Oct/Nov 2021

A mass hangs on a spring that is fastened to a stationary support, as shown in Fig. 3.1. The mass moves up and down in simple harmonic motion about its equilibrium position.

Oct/Nov 2022

A solid metal sphere with mass $0.81\,\text{kg}$ hangs on a string. It performs small side-to-side oscillations, as illustrated in Fig. 4.1. The sphere’s motion can be treated as simple harmonic, with amplitude $0.036\,\text{m}$ and period $3.0\,\text{s}$.

Oct/Nov 2023

A sphere made of heavy metal with mass $0.81\,\text{kg}$ hangs on a string. It moves in small side-to-side oscillations, as illustrated in Fig. 4.1. These oscillations can be treated as simple harmonic, with amplitude $0.036\,\text{m}$ and period $3.0\,\text{s}$.

Oct/Nov 2023