Periodic motion (3.6.1)
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For objects in circular motion, it is useful to work with angles measured in radians rather than degrees.
In the diagram below, the angle represented is equal to one radian when the arc length is the same length as the radius of the circle.
The angle in radians is found by:
Half the circumference is equal to the radius multiplied by Using the formula for the angle in radians gives:
Therefore, radians is equivalent to The whole circumference is equal to two times the radius multiplied by Again using the formula above gives:
Therefore, radians is equivalent to
To convert from radians to degrees, use:
To convert from degrees to radians, use:
These equations should be memorised. Expressing angles in radians as fractions of pi when possible is useful for maintaining precision. The diagrams below show some common conversions.
Calculators are normally set by default to measure angles in degrees when calculating trigonometric functions, but they will have an option to change from ‘degrees mode’ to ‘radians mode’.
The time period of an object in circular motion is the time it takes to make one complete rotation Time period is measured in seconds ().
The frequency of an object in circular motion is how many revolutions it completes in one second. Frequency is measured in hertz () or revolutions per second ().
The frequency and time period of an object in circular motion are inversely proportional:
As the time taken for a complete revolution decreases, the number of revolutions per second increases, and vice versa.
Question walkthrough
Finding time period after tripling frequency
Uses the inverse relationship f=1/T to find the new time period for a full rotation after the frequency of rotation triples.
In physics, quantities can either be scalar or vector. A scalar quantity has just magnitude, whereas a vector has both magnitude and direction.
- Angular speed is an example of a scalar quantity. The amount of radians per second an object is turning through can be measured, but the direction is not relevant.
- Angular velocity is a measurement of both the number of radians per second an object is turning through and the direction this rotation is occurring in. The direction will often be defined as being positive or negative, similar to what would happen when measuring linear velocity.
It is important to note that the direction of angular velocity will not be considered for your AQA A-level physics exam. However, it is a useful distinction to make for your understanding.
When calculating angular speed or angular velocity, the same formulae can be applied to both.
For a complete revolution, an object turns through an angle of in one time period, Therefore, the equation:
becomes:
Frequency is related to the time period by:
Thus, angular speed or angular velocity is also given by the equation:
Newton’s first law states that an object will continue travelling at a constant speed in a straight line unless acted upon by a net force.
An object following a circular path changes direction, and therefore, a force must act on it. This force is directed towards the centre of the circle, perpendicular to the object’s velocity.
A centripetal (centre-seeking) force keeps an object moving at constant speed in a circle, but causes the direction of the object’s motion to change.
While the speed of an object undergoing uniform circular motion remains constant, the constantly changing direction means the object has a changing velocity. It is important to note that speed is a scalar quantity and velocity is a vector.
A change in velocity means an object is accelerating. The centripetal force provides this acceleration. Newton’s second law states that force and acceleration are proportional to each other; you cannot have one without the other.
The centripetal force acts towards the centre of an object’s circular motion. Examples of centripetal forces are shown below:
If an object is moving in a circular path and the centripetal force is removed, then the object will fly off at a tangent.
An example of this is cutting the string attached to a ball being swung in a circle parallel to the ground.
Once the centripetal force is removed, Newton’s first law applies again – the object will travel in a straight line at a constant speed unless another resultant force acts upon it.
The centrifugal force is the name given to the fictitious outward pseudoforce you experience when you are turning.
For example, if you were sitting on the back seat of a car as it went round a corner you might feel like you are being pushed away from the centre of the turn and slide away from the centre of the turn.
However, this is just your inertia wanting you to continue on a straight path. When the car turns, the side of the bus pushes you towards the centre of the turn. This would be a real centripetal force caused by the reaction force between you and the side of the car.
Be careful not to confuse centripetal force with centrifugal force in questions. Exam questions will only require you to perform calculations involving centripetal force.
The linear speed of an object can be found by dividing the distance travelled by the time taken.
In the case of an object moving with circular motion, the speed can be found by dividing the circumference of the circle representing the object’s trajectory by the time taken to complete one full rotation (the period):
Where:
- is linear speed, measured in metres per second (),
- is the radius of the circle, measured in metres (),
- is the time period, measured in seconds ().
The angular speed of an object moving with circular motion is given by:
Where is the angular speed, measured in radians per second ().
Combining these two equations using substitution leads to the relationship between linear speed and angular speed :
The equation implies that the linear speed of an object moving with circular motion is proportional to the radius, if the angular speed is constant.
An example of this is two points at different distances from the hub on a bike wheel. They will both have the same angular speed because they will both take the same amount of time to complete one full revolution (the same time period).
However, their linear speeds will be different, as they have to travel different distances to complete one rotation.
In the diagram above, point A is further from the centre than point B and will need a greater linear speed in order to complete one full rotation in the same amount of time as point B.
Question walkthrough
Linear Speed of Objects on a Rotating Disc
Use the shared angular speed of two children on a merry-go-round to find one's linear speed from the other's.
As an object travels in a circular path, its direction is constantly changing. Therefore, its velocity must be constantly changing as velocity is a vector. If the velocity of an object is changing, the object is accelerating.
This acceleration is known as a centripetal acceleration and is directed towards the centre of the circle and perpendicular to the velocity.
The centripetal acceleration of an object moving with circular motion is given by:
Where:
- is the centripetal acceleration, measured in metres per second squared (),
- is the linear speed, measured in metres per second (),
- is the radius, measured in metres ().
Combining the equation above with the equation that links linear speed to angular speed produces an alternative way to find the centripetal acceleration :
combined with returns
Where is the angular speed, measured in radians per second ().
Question walkthrough
Comparing Centripetal Acceleration on a Turntable
Use the shared angular speed of two objects on a turntable to find one's centripetal acceleration from the other's.
In circular motion, the centripetal force is directed towards the centre of the circle and is also perpendicular to the velocity of the object. Newton’s second law states that force is proportional to acceleration:
Therefore, the equations for centripetal acceleration can be converted into equations for centripetal force by multiplying the acceleration by the mass of the object moving with circular motion. Equations for centripetal acceleration are:
So the equations for centripetal force are:
Where:
- is the mass of the object moving with circular motion, measured in kilograms (),
- is the linear speed, measured in metres per second (),
- is the radius, measured in metres (), and
- is the angular speed, measured in radians per second ().
Centripetal force it is not a force itself. It is a measure of the pull of another force towards the centre of a circle.
In circular motion, the centripetal force is provided by a force acting towards the centre, such as friction, gravity, tension, or a combination of forces.
An example of a combination of forces constituting centripetal force is a ball spun in a vertical circle on a string. The centripetal force () is a combination of the object’s weight () and the tension in the string ().
The weight of the object acts downwards and is constant (the mass does not change).
The centripetal force is also constant if the radius and speed stay constant (true for both linear and angular speed).
The tension in the string is the force that can change in size and direction:
- When the ball is at the bottom, the tension supports the weight and provides the centripetal force.
- When the ball is at the top, its weight points to the centre, providing some centripetal force; the tension provides the rest.
When calculating tension or weight, remember that the resultant centripetal force is always directed towards the centre.
Circular motion can be investigated using a whirling bung experiment:
- A rubber bung is attached to a thin piece of string, which is passed through a hollow glass tube.
- A weight is suspended from the other end of the string.
- The student holds the glass tube and whirls the rubber bung above their head horizontally with circular motion.
The suspended weight creates tension in the string as the bung is whirled, providing the centripetal force to keep it moving in a circular path.
Question walkthrough
Frictional Force for Cars on Different Radii
Calculate the frictional force needed for two cars at the same speed to take corners of different radii.
Question walkthrough
String Tension in Vertical Circular Motion
Calculate the tension in a string at the top and bottom of a vertical circle for a ball undergoing circular motion.
A simple pendulum and a mass on a spring are examples of simple harmonic oscillators (SHO). These objects oscillate with what is known as simple harmonic motion (SHM).
Simple harmonic motion occurs when an object oscillates to and fro, either side of an equilibrium position. The equilibrium is the position in which it a real oscillator would come to rest due to friction.
A restoring force tries to return the oscillator to equilibrium. The restoring force could be gravity, tension, or other forces. As shown in the diagram below, the size of the restoring force is directly proportional to the size of the displacement from equilibrium.
Newton’s second law states that the size of the force is directly proportional to the size of the acceleration. Therefore, when the restoring force towards the equilibrium is greatest, so is the acceleration towards the equilibrium, as shown in the diagram below.
The definition of an object moving with simple harmonic motion (SHM) has two key conditions:
An oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position, and is directed towards the equilibrium, expressed mathematically as:
Where:
- represents the acceleration measured in metres per second
- is the proportionality constant.
- is displacement from the equilibrium measured in metres
The negative sign represents that the acceleration is in the opposite direction to the displacement, as it is always directed towards the equilibrium.
In a proportional relationship, there is always a constant . In the SHM equation, the full equation is as follows:
The constant in this case is actually the angular frequency squared, and is measured in
Determining the frequency of a real object moving with simple harmonic motion requires measuring the time period.
Using the fiducial marker placed at the equilibrium as your reference for starting and stopping the stopwatch, one complete time period would look like the diagram below:
One time period is often short in practice, so to reduce uncertainty, you should measure at least ten time periods. Then divide the measurement of ten time periods by ten to find the value of one time period.
Once you have the time period, you can find the frequency using the formula below, making sure to enter the time period in seconds:
There are two equations you can use to calculate the displacement of an object at any time that is moving with simple harmonic motion.
The first equation or second equation below should be used if the oscillations begin at maximum displacent or the equilibrium positions respectively.
Where:
- is the object’s displacement from the equilibrium position. It is measured in metres.
- is the amplitude, which is the maximum displacement of the object from equilibrium, measured in metres.
- is the angular frequency, defined as how quickly an object completes one full oscillation (one time period).
- is the time since the oscillations started, measured in seconds.
Displacement and amplitude are vectors, so can be positive or negative. Moreover, the shorter the time period of the oscillation, the greater the angular frequency. The angular frequency remains constant throughout simple harmonic motion as the time period is constant.
It is important to note that your calculator should be in radians mode when using these equations.
In order to find the acceleration of an object at a certain displacement from the equilibrium, we can use the equation which is defined by the definition of simple harmonic motion.
In this equation,
- is the angular frequency and is measured in
- is acceleration measured in
- is the displacement from the equilibrium position measured in metres.
Question walkthrough
Finding SHM Displacement at Given Time
Uses the SHM displacement equation x=Acos(ωt) to find a pendulum's displacement at a specific time, given its amplitude and frequency.
Question walkthrough
SHM Acceleration from Displacement and Period
Finds the time period from an oscillation count, then uses the SHM acceleration equation to find a pendulum's acceleration at a given displacement.
The velocity of an object moving with simple harmonic motion at a particular displacement from the equilibrium can be calculated using this equation:
In this equation:
- is velocity measured in metres per second
- is the angular frequency and is measured in radians per second
- is the amplitude measured in metres
- is the displacement from the equilibrium measured in metres
The reason for the is that an object can pass through the equilibrium from both directions with positive or negative velocity.
An object moving with simple harmonic motion has its maximum velocity at the equilibrium, when :
Therefore, the equation for the velocity of an object moving with SHM can be simplified to this:
Question walkthrough
Finding Speed in Mass-Spring SHM
Uses the SHM speed equation to find a mass-spring system's speed at a given displacement and at equilibrium, from its amplitude and time period.
A pendulum moves with simple harmonic motion. The graph below shows how the pendulum’s displacement would be represented graphically over one time period, with the pendulum initially displaced to the left of the equilibrium with positive amplitude.
- Initially the pendulum bob starts at the positive amplitude.
- Quarter of a time period, the pendulum bob passes through the equilibrium, so the displacement equals zero.
- Half a time period the pendulum bob is now at the negative amplitude
- Three-quarters of a time period, the pendulum bob is passing back through the equilibrium in the opposite direction.
- Full time period the pendulum bob is now back at its starting position, which is the positive amplitude.
A pendulum moves with simple harmonic motion. The graph below shows how the velocity of the pendulum would be represented graphically over one time period. Starting with the pendulum displaced to the left of the equilibrium with positive amplitude.
- Initially the pendulum bob starts at the positive amplitude. At either amplitude, the velocity of the bob is momentarily zero.
- Quarter of a time period the bob passes through equilibrium where the velocity is maximum. The bob moves towards the negative amplitude with negative velocity.
- Half a time period the bob is at the negative amplitude. The velocity is momentarily zero.
- Three-quarters of a time period the bob moves through the equilibrium in the opposite direction. It has maximum positive velocity.
- Full time period the bob returns to its starting position; both the positive amplitude and velocity are zero.
A pendulum moves with simple harmonic motion. The graph shows a graphical representation of the acceleration of a pendulum over one time period. We begin with the pendulum displaced to the left (which we are indicating as the positive amplitude).
- Initially the restoring force is towards the right (negative side). Force and acceleration are proportional to displacement, so acceleration is greatest at the amplitude.
- Quarter of a time period the pendulum bob passes through equilibrium. Acceleration is proportional to displacement from the equilibrium, so at equilibrium, acceleration equals zero.
- Half a time period the bob is now at the negative amplitude. The restoring force and acceleration are greatest and directed towards the positive side.
- Three-quarters of a time period the bob passes through the equilibrium in the opposite direction, so acceleration is zero again.
- Full time period the bob returns to its starting position and the positive amplitude and acceleration are greatest in the negative direction.
The diagram below shows a pendulum undergoing simple harmonic motion. The displacement, velocity and acceleration of the bob are shown together over a full time period, so you can observe its motion and analyse the phase difference between them graphically.
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The definition of simple harmonic motion states that the acceleration of an object moving with simple harmonic motion is directly proportional to the displacement from the equilibrium and is always directed towards the equilibrium position.
The graph shows how an object’s acceleration varies with displacement; it is represented by a straight line that passes through the origin.
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- When the object is at its maximum negative amplitude, it will experience the greatest positive acceleration.
- When at the equilibrium, it will experience no acceleration.
- When at the maximum positive amplitude, it will experience the greatest negative acceleration.
An object moving with simple harmonic motion has the greatest velocity when it passes through the equilibrium position. However, the sign of the velocity changes depending on the direction of travel.
When the object is at either maximum amplitude, its velocity is zero, representing the moment it changes direction.
Shown graphically, these four points are joined by a circle.
An oscillator moving with simple harmonic motion is known as an isochronous oscillator. Isochronous oscillations mean that the time it takes to complete one oscillation is independent of its amplitude, for example, no matter how much you initially displace a pendulum or a mass on a spring; its time period remains the same.
The time period of a simple pendulum in seconds can be calculated using the equation below:
The equation shows that the only variables that can change the time period are:
- the length of the pendulum which is measured from the point of suspension to the centre of mass of the bob, the units will be metres and
- the acceleration due to gravity measured in metres per second per second
The time period of a mass on a spring in seconds can be calculated using the equation below:
The only variables that can change the time period of an oscillation are:
- the mass, measured in kilograms and
- the spring constant, measured in newtons per metre
Question walkthrough
Finding Pendulum Length from Time Period
Rearranges the simple pendulum time period formula to find its length, testing that the period is independent of amplitude for SHM.
The energy of an oscillator performing simple harmonic motion is continually transferred between potential energy and kinetic energy.
The potential energy of an oscillator can come in different forms, for example:
- Gravitational potential energy for a pendulum bob when it is higher than the lowest point of its swing
- Elastic potential energy for a mass on a spring when the spring is compressed or stretched.
The kinetic energy of an object is given as:
where:
- is the kinetic energy measured in joules ()
- is the mass measured in kilograms ()
- is the velocity measured in metres per second ().
In simple harmonic motion:
- At the equilibrium position, the speed of the oscillator is maximum and, therefore, this is where the kinetic energy is also greatest.
- At the maximum displacement (equal to the amplitude), the velocity is zero; therefore, so is the kinetic energy.
The graph below illustrates the variation in kinetic energy of a pendulum bob over two distinct time periods.
The maximum kinetic energy of the system can be found using the equation:
The maximum velocity of an object moving with simple harmonic motion in is:
Where:
- is the angular velocity measured in radians per second ()
- is the amplitude measured in meters ().
The potential energy of an object in simple harmonic motion depends on the displacement from the equilibrium position:
- At equilibrium, the potential energy of the oscillator is zero.
- At maximum displacement, the potential energy is maximum.
The graph below illustrates the variation in potential energy of the oscillator over two time periods
If the potential energy of the oscillator is in the form of gravitational potential energy, then we can use the following equation:
where:
- is the height above the equilibrium and is measured in metres ()
- is the mass measured in kilograms ()
- is acceleration due to gravity measured in metres per second squared ().
If the potential energy is in the form of elastic potential energy, then we can use the following equation:
where:
- is the stiffness constant of the object and is measured in Newtons per metre ()
- is the change in length of the object and is measured in metres ().
The maximum elastic potential energy will be when the change in length is equal to the maximum displacement (amplitude), so the equation becomes:
where:
- is the amplitude measured in meters ().
The total energy of a simple harmonic oscillator is the sum of its kinetic energy and potential energy.
The total energy remains constant unless an external force, such as friction, acts on the system which causes a damping effect.
An example of this is a pendulum bob undergoing simple harmonic motion:
- At the top of the swing, the bob is stationary and all of its energy is in the form of potential energy.
- As the bob descends, it gains speed and hence kinetic energy, but loses potential energy as its height decreases.
- At the equilibrium position, all of the bob’s energy is in the form of kinetic energy.
- As the bob rises again, it loses speed and, hence, kinetic energy, but gains potential energy again.
To calculate the total energy of the system, it is easier to calculate the maximum value of the kinetic energy or the potential energy than the sum of the kinetic energy and potential energy at some point between the equilibrium and the amplitude:
Question walkthrough
Time Period from Total Energy in SHM
Find the time period of an oscillator from its mass, amplitude, and total energy using v_max and angular frequency.
As an object moving with simple harmonic motion oscillates between two amplitudes, it constantly exchanges energy between kinetic and potential forms.
- The kinetic energy is greatest at the equilibrium, where the object is moving fastest and zero at the amplitude.
- The potential energy is greatest at the amplitudes and zero at the equilibrium.
- The total energy is the sum of the kinetic and potential energies at any particular point and is a constant as long as no external forces are acting, such as friction.
Shown below is a graph demonstrating how the different types of energy vary with displacement from the equilibrium position.
Energy changes for an ideal simple pendulum over time, where is the displacement from the equilibrium position:
Energy changes for an ideal mass spring system over time, where is the displacement from the equilibrium position:
Question walkthrough
Speed at Equal KE and PE in SHM
Calculate the speed of a pendulum bob at the point in its oscillation where kinetic and potential energy are equal.
Free oscillations occur when an object oscillates with no transfer of energy either to or from the surroundings. In a free oscillation, there are no external forces acting on the object oscillating.
The frequency of a free oscillation is known as the resonant (natural) frequency of the oscillator.
In practice, there are very few examples of free oscillations due to damping effects, such as air resistance and friction. The vibrations of particles in a gas, however, are considered to be free oscillations.
Forced oscillations occur when a periodic external force acts on an object that is already oscillating.
The frequency of the external force is referred to as the driving frequency.
The external force provides energy to overcome the losses caused by damping.
An example of a forced oscillation is a child being pushed on a swing by another person.
The child is given a push at the start of each swing to maintain the swinging motion and counteract the damping forces.
In real-life situations, an oscillating object experiences resistive forces, which act in the opposite direction to the velocity of the object.
Examples of resistive forces:
- Air resistance is experienced by an object as it moves through air.
- Drag is experienced by an object moving through water (or any fluid).
- Friction is experienced by an object moving along a surface.
Resistive forces cause the amplitude and oscillation to decrease due to energy being transferred away from the oscillator. This is known as damping.
Although the amplitude of an oscillator decreases when it experiences damping, the frequency of the oscillations remains constant, as long as the damping force is not so large that it stops the oscillation completely.
For example, pendulum clocks still tell the time accurately as the swings of the pendulum decrease.
There are three types of damping that an oscillator can experience. One of these is light damping:
- The size of the resistive force is small.
- The amplitude size decays over time.
- The frequency of oscillation remains constant.
An example of light damping is a real pendulum’s bob as it swings through the air.
There are three types of damping that an oscillator can experience. One of these is heavy damping:
- The size of the resistive force is large.
- Similar to light damping, but the amplitude decays exponentially more quickly over time.
- The frequency of oscillation remains constant.
An example of heavy damping is the suspension on a rough terrain vehicle, which dissipates energy quickly so the oscillations die away rapidly and the vehicle does not continue to bounce after going over a bump.
There are three types of damping that an oscillator can experience. One of these is critical damping:
- A critically damped oscillator will return to the equilibrium position almost immediately after being initially displaced.
- The damping force is great enough to prevent oscillations, in contrast to both light and heavy damping.
An example of critical damping is a slow-closing, heavy door that incorporates a damping mechanism to prevent it from slamming shut.
Question walkthrough
Exponential Amplitude Decay of a Damped Pendulum
Use the constant ratio of successive amplitudes in a damped pendulum to predict the amplitude after further oscillations.
Oscillators have a natural frequency.
An object oscillates at its natural frequency when performing a free oscillation, where no energy is being transferred to or from the surroundings.
For example, if a guitar string is plucked once, it will vibrate at its natural frequency. The notes from the different strings on a guitar correspond to their natural frequencies, which depend on their thickness and tension.
In a forced oscillation, an external force is applied periodically. This is known as the driving frequency.
When the driving frequency of the external force matches the natural frequency of the oscillator, then a phenomenon called resonance occurs.
During resonance, the energy transfer from the driving force to the oscillator is most efficient. This causes the amplitude of the oscillator to increase quickly to its maximum.
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An example of resonance is a child being pushed on a swing.
The child on the swing acts like a pendulum, with a natural frequency which depends on the length of the swing.
The adult applies a force at regular intervals. If the frequency of the driving force applied matches the natural frequency of the swing, then the amplitude of the oscillations will dramatically increase.
The amplitude of an object oscillating at its natural frequency varies with driving frequency.
When the driving frequency is below the natural frequency, the amplitude of oscillations is small.
As the driving frequency approaches the natural frequency of the oscillator, the amplitude increases rapidly, reaching a maximum when the driving frequency equals the natural frequency; this is known as resonance.
When the driving frequency exceeds the natural frequency, the amplitude begins to decrease again.
If a system undergoing a forced oscillation experiences damping, the amplitude of the oscillations decreases.
The greater the damping, the greater the decrease in amplitude at all frequencies.
For a resonating system, as the damping force increases, the resonance peak decreases and becomes broader. In addition, the peak shifts to a lower driving frequency.
Question walkthrough
Sketching Resonance Curves for Increasing Damping
Sketch how the amplitude-frequency resonance curve changes as damping is progressively increased, labelling the natural frequency.
A Barton’s pendulum is a system used to display the effect of resonance.
A series of pendulums of different lengths is suspended from the same horizontal, flexible string.
One of the pendulums (X) is much heavier than the others. Pendulum X is displaced and begins moving with simple harmonic motion, with a frequency determined by its length.
The flexible string feels a force from the motion of pendulum X, and it vibrates at the same frequency. The other pendulums start to oscillate in response.
Pendulum D will oscillate with the greatest amplitude because it is the same length as pendulum X, so they will both have the same natural frequency.
Pendulum X provides an external periodic driving frequency which matches the natural frequency of pendulum D, causing it to resonate. The other pendulums will oscillate, but with a smaller amplitude than pendulum D.
Stringed instruments display the effect of resonance. When a string is plucked, it vibrates and stationary waves are formed. Their frequencies are the resonant frequencies of the string.
The stationary waves consist of a series of nodes and antinodes. Their wavelengths are determined by the length of the string.
The string vibrates at several of these resonance frequencies simultaneously. The higher frequencies, known as harmonics, determine the sound of the instrument.
However, these vibrations move very little air and produce almost no sound on their own. The body of the guitar is designed to have a natural frequency similar to the frequency of the stationary waves.
The body of the guitar resonates when the strings vibrate at the same frequency, causing the larger body to vibrate and displace a significantly greater amount of air, thereby creating louder sound waves.