Simple harmonic oscillations

A mass is moving in simple harmonic motion with amplitude $x_0$. The greatest magnitude of the mass’s velocity is $v_0$. On Fig. 3.1, indicate how the velocity $v$ of the mass varies with displacement $x$.

Feb/March 2018

A tube with a circular cross-section, closed at one end, has area $A$ and holds sand. The combined mass of the tube and the sand is $M$. As shown in Fig. 3.1, the tube floats vertically in a liquid of density $\rho$. It is pressed down a small distance into the liquid and then let go.

Feb/March 2019

The defining equation of simple harmonic motion is $a = -\omega^2 x$. State what the minus $(-)$ sign indicates in the equation.

Feb/March 2021

A small wooden cuboid block of mass $m$ floats in water, as illustrated in Fig. 3.1. Its top face is horizontal and has area $A$. The density of the water is $\rho$.

Feb/March 2022

State what oscillations mean.

May/June 2010

Define what is meant by simple harmonic motion.

May/June 2011

A ball is kept between two fixed points A and B by two stretched springs, as shown in Fig. 3.1. The ball can oscillate along the straight line AB. The springs stay stretched, and the motion of the ball is simple harmonic. Fig. 3.2 shows how the displacement $x$ of the ball from its equilibrium position changes with time $t$.

May/June 2013

A ball is trapped between two fixed points A and B by means of two stretched springs, as illustrated in Fig. 3.1. It is free to move to and fro along the straight line AB. Because the springs stay stretched, the ball undergoes simple harmonic motion. Fig. 3.2 shows how the displacement $x$ of the ball from its equilibrium position varies with time $t$.

May/June 2013

An oscillator causes a metal plate to vibrate up and down, as illustrated in Fig. 2.1. Sand is then scattered onto the plate. Fig. 2.2 shows how the sand’s acceleration $a$ varies with displacement $y$.

May/June 2018

An oscillator makes a metal plate vibrate vertically, as illustrated in Fig. 2.1. A little sand is then sprinkled onto the plate. Fig. 2.2 shows how the sand’s acceleration, $a$, changes with displacement $y$.

May/June 2018

A hollow tube, closed at one end, has a cross-sectional area $A$ of $24\,\text{cm}^2$. Sand inside the tube makes the combined mass $M$ of the tube and sand $0.23\,\text{kg}$. The tube floats vertically in a liquid of density $\rho$, as shown in Fig. 3.1. The depth of the tube’s bottom below the liquid surface is $h$. The tube is moved vertically and then let go. Fig. 3.2 shows how the depth $h$ varies with time $t$.

May/June 2019

A spring is attached vertically to a fixed support at one end. A mass $M$ is attached to the other end of the spring, as shown in Fig. 3.1. The mass is pulled downward and then let go. It then undergoes simple harmonic motion. Fig. 3.2 shows how the spring length $L$ varies with time $t$.

May/June 2019

A hollow tube, sealed at one end, has a cross-sectional area $A$ of $24\,\text{cm}^2$. Sand is added to the tube so that the combined mass $M$ of the tube and the sand is $0.23\,\text{kg}$. The tube floats upright in a liquid of density $\rho$, as shown in Fig. 3.1. The depth of the bottom of the tube below the liquid surface is $h$. The tube is moved vertically and then released. Fig. 3.2 shows how the depth $h$ changes with time $t$.

May/June 2019

The piston inside the cylinder of a car engine undergoes simple harmonic motion. It travels from the highest point in the cylinder to the lowest point, as shown in Fig. 3.1. The separation moved by the piston between the positions displayed in Fig. 3.1 is $9.8\,\text{cm}$. The mass of the piston is $640\,\text{g}$. At one chosen engine speed, the piston makes $2700$ oscillations in $1.0\,\text{minute}$.

May/June 2020

This dish is formed from part of a hollow glass sphere. It is secured to a horizontal table and holds a small solid ball of mass $45\,\text{g}$, as shown in Fig. 4.1. The ball’s horizontal displacement from the centre $C$ of the dish is $x$. At the beginning, the ball is kept at rest with $x = 3.0\,\text{cm}$. It is then released. Fig. 4.2 shows how the horizontal displacement $x$ of the ball from point $C$ changes with time $t$. The ball’s motion in the dish is simple harmonic, with acceleration $a$ given by $a = -\left(\frac{g}{R}\right)x$, where $g$ is the acceleration of free fall and $R$ is a constant that depends on the dimensions of the dish and the ball.

May/June 2020

A piston inside a car engine cylinder undergoes simple harmonic motion as it travels within the cylinder. It moves from the highest position in the cylinder to the lowest position, as shown in Fig. 3.1. The separation between the two positions in Fig. 3.1 is $9.8\ \text{cm}$. The piston has a mass of $640\ \text{g}$. At a certain engine speed, the piston makes $2700$ oscillations in $1.0$ minute.

May/June 2020

A pendulum is made from a bob (a small metal sphere) hung from the lower end of a length of string. The upper end of the string is fixed to a point. The bob makes small oscillations about its equilibrium position, as shown in Fig. 4.1. The length $L$ of the pendulum, taken from the fixed point to the centre of the bob, is $1.24\,\text{m}$. The acceleration $a$ of the bob changes with its displacement $x$ from the equilibrium position, as shown in Fig. 4.2.

May/June 2022

A pendulum is formed by a bob (tiny metal sphere) hanging from one end of a string, with the other end fixed at a point. The bob makes small oscillations about its equilibrium position, as shown in Fig. 4.1. The pendulum length $L$, measured from the fixed point to the centre of the bob, is $1.24\,\text{m}$. The acceleration $a$ of the bob changes with its displacement $x$ from equilibrium as shown in Fig. 4.2.

May/June 2022

A block with mass $m$ vibrates vertically on a spring, as illustrated in Fig. 4.1. The block’s acceleration $a$ changes with displacement $x$ from equilibrium, as plotted in Fig. 4.2. The oscillation amplitude is $3Y$ and the greatest acceleration is $2A$.

May/June 2024

A block is attached to a spring and hangs freely. It oscillates vertically in simple harmonic motion. The block's velocity $v$ varies with time $t$ according to $v = 0.56\cos 16t$, where $v$ is in $\text{m s}^{-1}$ and $t$ is in $\text{s}$.

May/June 2025

State the meaning of simple harmonic motion.

May/June 2025

Figure 3.1 shows a cylinder and piston from a car engine. The piston’s vertical motion inside the cylinder is taken to be simple harmonic. When the piston is at its lowest point, its top surface is at AB; when it reaches its highest point, the top surface is at CD, as indicated in Fig. 3.1.

Oct/Nov 2010

A ball is trapped between two fixed points A and B by two stretched springs, as shown in Fig. 4.1. The ball can oscillate horizontally along AB. While it moves, both springs stay stretched and never go past their limits of proportionality. Fig. 4.2 shows how the acceleration $a$ of the ball varies with its displacement $x$ from the equilibrium position.

Oct/Nov 2012

A ball is trapped between two fixed points A and B by two stretched springs, as shown in Fig. 4.1. It can oscillate freely in the horizontal direction along AB. While it moves, the springs stay stretched and never go beyond their proportionality limit. Fig. 4.2 shows how the acceleration $a$ of the ball varies with its displacement $x$ from equilibrium.

Oct/Nov 2012

A trolley of negligible mass that moves without friction is joined to a fixed point $A$ by a spring. A second spring connects the trolley to a variable frequency oscillator, as shown in Fig. 2.1. Both springs stay extended, but still within the limit of proportionality. At the start, the oscillator is switched off. The trolley is moved horizontally along the line joining the two springs and then let go. Fig. 2.2 shows how the trolley’s velocity $v$ varies with time $t$.

Oct/Nov 2012

State what simple harmonic motion means.

Oct/Nov 2014

State the meaning of simple harmonic motion.

Oct/Nov 2014

State the meaning of simple harmonic motion.

Oct/Nov 2014

State what is meant by simple harmonic motion.

Oct/Nov 2015

To illustrate simple harmonic motion, a student connects a trolley to two identical stretched springs, as shown in Fig. 3.1. The trolley has mass $m$ of $810\,\text{g}$. It is pulled along the line of the two springs and then let go. Its resulting acceleration $a$ is described by the expression $a = -\frac{2kx}{m}$ where the spring constant $k$ for each spring is $64\,\text{N m}^{-1}$ and $x$ represents the trolley’s displacement.

Oct/Nov 2016

As shown in Fig. 4.1, a mass is suspended vertically from a fixed point by a spring. The mass is then displaced vertically and released. Its resulting oscillations are simple harmonic. Fig. 4.2 shows how the spring length $l$ changes with time $t$.

Oct/Nov 2016

To illustrate simple harmonic motion, a student connects a trolley to two identical stretched springs, as shown in Fig. 3.1. The trolley has mass $m$ of $810\,\text{g}$. It is pulled along the direction of the two springs and then let go. The resulting acceleration $a$ of the trolley is described by $a = -\frac{2kx}{m}$, where the spring constant $k$ for each spring is $64\,\text{N m}^{-1}$ and $x$ is the trolley’s displacement.

Oct/Nov 2016

State, in terms of simple harmonic motion, what angular frequency means.

Oct/Nov 2017

Define the radian in relation to a circle.

Oct/Nov 2017

State, with reference to simple harmonic motion, what angular frequency means.

Oct/Nov 2017

A mass is hanging vertically from a fixed point by a spring, as shown in Fig. 4.1. The mass moves up and down. Fig. 4.2 shows how the acceleration $a$ changes with displacement $x$ of the mass.

Oct/Nov 2019

A mass hangs vertically from a fixed point on a spring, as shown in Fig. 4.1. The mass is moving in vertical oscillation. Fig. 4.2 shows how the acceleration $a$ of the mass varies with displacement $x$.

Oct/Nov 2019

As shown in Fig. 3.1, a pendulum is made from a metal sphere P hanging from a fixed point by a thread. The centre of gravity of sphere P is at a distance $L$ from the fixed point. The sphere is pulled to one side and then let go so that it oscillates. It may be assumed to execute simple harmonic motion.

Oct/Nov 2020

A pendulum is formed by a metal sphere P hanging from a fixed point on a thread, as shown in Fig. 3.1. The centre of gravity of sphere P is a distance $L$ from the fixed point. The sphere is displaced to one side and then let go, so that it oscillates. It may be assumed to oscillate with simple harmonic motion.

Oct/Nov 2020

A trolley moving on a smooth surface is connected to fixed blocks by springs, as illustrated in Fig. 4.1. It undergoes horizontal oscillations about its equilibrium position with amplitude $12\,\text{cm}$. Fig. 4.2 shows how the trolley’s acceleration $a$ varies with displacement $x$ from equilibrium. Friction between the trolley and the surface may be taken as negligible.

Oct/Nov 2021

An object hangs from a spring fixed at one end, as illustrated in Fig. 3.1. The object moves up and down in simple harmonic motion about its equilibrium position.

Oct/Nov 2022

State the meaning of simple harmonic motion.

Oct/Nov 2024

State the meaning of simple harmonic motion.

Oct/Nov 2024

A steel ball at the lower end of a fine string makes small oscillations, as illustrated in Fig. 5.1. The displacement of the ball’s centre from equilibrium is $x$. Fig. 5.2 plots how the ball’s acceleration $a$ varies with $x$.

Oct/Nov 2025

Describe the motion of an object in uniform circular motion in terms of velocity and acceleration.

Oct/Nov 2025

An object is undergoing oscillation. Fig. 4.1 presents how the acceleration $a$ of the object changes with displacement $x$ measured from the equilibrium position. Fig. 4.2 presents how the kinetic energy $E_K$ of the object varies with time $t$.

Oct/Nov 2025