OCR A Level Physics - Oscillations, Chapter 17

In SHM, what is Displacement?

Displacement x is the distance from the equilibrium position. It is measured in metres.

In SHM, what is Amplitude?

Amplitude A is the maximum displacement from the equilibrium position. It is measured in metres.

In SHM, what is the Time Period?

The time period T is the time taken to complete one full oscillation. It is measured in seconds.

In SHM, what is Frequency?

Frequency f is the number of complete oscillations per unit time. It is measured in Hertz.

In SHM, what is Phase Differernce?

Phase Difference is the difference in displacement between two oscillating objects. It is measured in radians or degrees.

In SHM, what is Angular Frequency?

Angular frequency is the rate of change of angular position. It is given by the equation ω = 2π/T or ω =2πf where T is the time period and f is the frequency.

How do the Speed and Displacement Change as an object is Displaced from its Equilibrium and Released?

As the object is released, it will travel towards the equilibrium position at increasing speed. It reaches a max speed at the equilibrium and then slows down once it has gone past and eventually reaches maximum displacement (amplitude). It will then return to the equilibrium position, speeding up, and once more slows down to a stop when it reaches maximum negative displacement. This motion is repeated.

What is Simple Harmonic Motion?

Simple Harmonic Motion is a type of oscillation, where the acceleration of the oscillator is directly proportional to the displacement from the equilibrium position and acts toward the equilibrium position.

What is the Equation for Simple Harmonic Motion?

Simple harmonic motion is given by the equation a = - ω^2x, where a is the acceleration of the oscillator, ω is the angular frequency, and x is the displacement of the oscillator.

What Two features of all objects moving with SHM are shown using the equation a = - ω^2x?

In SHM, ω^2 is a constant so it shows that the acceleration a of the object is directly proportional to its displacement x, that is a ∝ x. The negative sign means that the acceleration of the object acts in the direction opposite to the displacement, returning the object to the equilibrium position.

What does a Graph of Acceleration against Displacement show for any object moving in SHM?

An acceleration against displacement graph is a straight line cutting through the origin since acceleration and displacement can be negative. The gradient of this line is equal to -ω^2, where ω is the angular frequency.

Why is the Constant Gradient, -ω^2, on a Graph of Acceleration against Displacement Useful?

The gradient equalling -ω^2 and being constant implies that the frequency and time period of an oscillator is also constant.

Why are Oscillators in SHM considered isochronous oscillators?

Oscillators in SHM are considered isochronous oscillators as the Time Period T of an oscillator is independent of the Amplitude A of the oscillator. This is because as amplitude increases so does the average speed of the swing, and so the time period doesn't change.

How does the Equation for SHM change for objects at Maximum Acceleration?

The equation a = - ω^2x will become amax = -ω^2A, where amax is the maximum acceleration, ω is the angular frequency, and A is the maximum amplitude. This is because the maximum acceleration occurs at the maximum amplitude.

How can we determine the Period and Frequency of objects moving with SHM?

1. Set the oscillator (such as a pendulum or a mass on a spring) into motion.

2. Use a stopwatch to measure the time taken for 10 oscillations to take place and divide this value by 10 to find the time period. This will reduce both random and human error. Since an oscillator in SHM is isochronous, the time period of the oscillation is independent of the amplitude.

3. A fiducial marker is used as the point to start and stop timings and can be placed at the equilibrium position to reduce uncertainty as the object is moving at its highest speed and because the object will continue to move through the equilibrium, even if the motion is damped.

4. Since frequency f is the reciprocal of the time period T, it can be calculated using the equation f = 1/T.

Investigate the Factors affecting SHM using a Pendulum? 1/2 PAG

1. Attach a ball bearing to a string and then attach this to the clamp stand.

2. Adjust the length l, which is from where the string is attached to the clamp stand to the centre of the ball bearing until it is 1m using a metre ruler.

3. Wait til the pendulum bob stops moving altogether, then place a fiducial marker directly underneath the bob. This represents the centre of oscillations and makes it easier to count how many oscillations the pendulum has undergone, reducing uncertainty.

4. Pull the pendulum bob to the side slightly and let it go so that it is oscillating with a small amplitude in a straight line. It has to be less than 10° otherwise it won't undergo SHM.

5. As the pendulum passes the fiducial marker, start the stopwatch and count the time taken for it to complete 10 full oscillations.

6. Take two more readings of the time period for 10 oscillations and calculate a mean, reducing random error.

7. Reduce the length l by 0.1m and repeat the last 3 steps.

8. Repeat the last step until the length l is 0.2m

Investigate the Factors affecting SHM using a Pendulum? 2/2 PAG

9. Divide the mean values of the time period at each length by 10 to get the time period of a single oscillation T.

10. Plot a graph of T^2 against l, and draw a line of best fit. It should be a straight line through the origin showing that l is directly proportional to T^2.

11. If T^2 is y and l is x, we can use the equation for SHM to find what our gradient represents: T^2 = (4π^2/g) x l so our gradient is equal to 4π^2/g so we can multiply it by 1/4π^2 to calculate g (acceleration of free fall).

12. We could use a light gate connected to a data logger to reduce uncertainty further.

Investigate the Factors affecting SHM using a Mass-Spring System? 1/2 PAG

1. Attach a spring to a clamp stand and attach a mass holder to the spring.

2. Wait until the spring stops moving altogether, then place a fiducial marker at the bottom of the mass holder, using the metre ruler to align it. This represents the centre of oscillations and makes it easier to count how many oscillations the mass-spring system has undergone, reducing uncertainty.

3. Pull the spring down slightly and let it go so that it's oscillating with a small amplitude and in a straight line.

4. As the bottom of the mass holder passes the fiducial marker, start a stopwatch and count the time taken for it to complete 10 full oscillations.

5. Take two more readings of the time period for 10 oscillations and calculate a mean, reducing random error.

6. Add a 100g mass to the mass holder and repeat the last 3 steps.

7. Repeat the last step until the total mass is 800g including the mass holder which is 100g.

Investigate the Factors affecting SHM using a Mass-Spring System? 2/2 PAG

8. Divide the mean values of the time period at each length by 10 to get the time period of a single oscillation T.

9. Plot a graph of T^2 against m and draw a line of best fit. It should be a straight line through the origin, showing that m is directly proportional to T^2.

10. If T^2 is y and m is x, we can use the equation for SHM to find what the gradient represents: T^2 = (4π^2/k) x m so our gradient is equal to 4π^2/k so we can multiply it by 1/4π^2 to calculate a value of k (the spring constant).

11. Wear eye protection with springs and add a counterweight to the base of the clamp to stop it from falling over.

12. We could use a light gate connected to a data logger to reduce uncertainty further.

What does the Displacement show in a Displacement-Time graph for an object going through SHM?

At zero displacement the object is at or moving through, its equilibrium position. At maximum displacement (amplitude), the object is at the top of its swing.

How can the Velocity be deduced from a Displacement-Time Graph?

The gradient of a displacement-time graph is equal to the velocity of the oscillator. At maximum displacements, the velocity is zero because the gradient of the graph is zero. The object in SHM has momentarily stopped, before returning to its equilibrium position. The velocity and the gradient of the graph are at a maximum as the pendulum moves through its equilibrium position.

If given a Displacement-Time graph, how do we translate this to get a Velocity-Time graph?

To get a velocity-time graph from a displacement-time graph, just shift the displacement-time graph by 90° to the left.

How can the Acceleration be deduced from a Velocity-Time graph?

The gradient of a velocity-time graph is equal to the acceleration of the oscillator. At maximum velocities, the acceleration is zero because the gradient of the graph is zero. The oscillator has gone through the equilibrium point. The acceleration and the gradient of the graph are at a maximum as the pendulum reaches the maximum displacement (amplitude).

If given a Displacement-Time or Velocity-Time graph, how do we translate this to get an Acceleration-Time graph?

The acceleration-time graph is inverted to the displacement-time graph and so the get an acceleration-time graph, we simply shift the displacement-time graph by 180° in either direction. This shows how acceleration is inversely proportional to displacement. To get an acceleration-time graph from a velocity-time graph, just shift the velocity-time graph by 90° to the left.

What Equation should you use to find the displacement of a SHM oscillator when it Begins oscillating from its Amplitude?

To find the displacement of a SHM oscillator beginning at its amplitude, we use the equation x = Acosωt, where x is the displacement, A is the amplitude, ω is the angular frequency, and t is the time.

What Equation should you use to find the displacement of a SHM oscillator when it Begins oscillating from its equilibrium?

To find the displacement of a SHM oscillator beginning at its equilibrium, we use the equation x = Asinωt, where x is the displacement, A is the amplitude, ω is the angular frequency, and t is the time.

What two things affect the Velocity of an oscillator in SHM?

When angular frequency increases but there is no change to amplitude, the oscillator will travel the same distance in a shorter time interval and so the maximum velocity of the oscillator, when displacement is zero, will increase. This would make the gradient of a displacement-time graph increase. Increasing the amplitude will also increase the velocity, as the oscillator is isochronous, so will travel a greater distance at the same time interval. To do this it must increase its velocity.

What is the Equation for the Velocity of an oscillator in SHM?

The velocity v of the oscillator is given by the equation v = ±ω x srqt(A^2 - x^2) where ω is the angular frequency, A is the amplitude, and x is the current displacement.

How can we Derive the Equation for the Max Velocity using the equation v = ±ω x srqt(A^2 - x^2)?

The maximum velocity occurs at the equilibrium position of an oscillator in SHM, where x = 0, so we can derive the equation Vmax = ωA to determine the maximum velocity of an oscillator.

How is Energy transferred in a Pendulum going through SHM?

For any object moving in SHM, the total energy remains constant, as long as there are no losses due to frictional forces. At the amplitude, the pendulum is briefly stationary and so has zero kinetic energy. Instead, all its energy is in the form of gravitational potential energy. As the pendulum falls, it loses potential energy and gains kinetic energy. As it moves through its equilibrium position, it has maximum velocity and so maximum kinetic energy. It will have no potential energy.

How is Energy transferred in a Mass-Spring System going through SHM?

For any object moving in SHM, the total energy remains constant, as long as there are no losses due to frictional forces. At the amplitude, the pendulum is briefly stationary and so has zero kinetic energy and instead has energy in the form of both gravitational potential and elastic potential if the mass is oscillating vertically. If it's oscillating horizontally, it will only have elastic potential energy. As the spring compresses, it loses its potential energy and gains kinetic energy. As it moves through its equilibrium position, it has maximum velocity and so maximum kinetic energy. It will have no potential energy.

What does a graph of Energy against Displacement show?

A graph of energy against displacement shows how the total energy of an oscillating system remains unchanged. There is a continuous interchange between potential energy and kinetic energy, but the sum at each displacement is always constant and equal to the total energy of the object. At the amplitude, the kinetic energy is zero whereas the potential energy is a maximum. At the equilibrium, the potential is zero whereas the kinetic energy is at a maximum.

How do we find the Elastic Potential Energy for a Mass-Spring system at its Amplitude on a horizontal track?

When the displacement x is equal to the amplitude A, the system will be stationary for an instant and so will have no kinetic energy. Therefore the total energy of the oscillator must be Ep = (1/2)kA^2, where k is the force constant.

How do we find the Kinetic Potential Energy for a Mass-Spring system on a horizontal track?

The kinetic energy Ek of the spring at any instant must be the difference between the total energy and the elastic potential energy Ep. Therefore, Ek = (1/2)kA^2 - (1/2)kx^2 = (1/2)k(A^2 - x^2).

What is Dampening?

Dampening is when an external force acts on an oscillator, reducing the amplitude of the oscillations over time.

What happens when there is Light Dampening?

When there is light dampening, the external damping force is small and the amplitude gradually decreases over time, but the period of the oscillations is almost unchanged. An example of light dampening is a pendulum oscillating in air.

What happens when there is Heavy Dampening?

When there is heavy dampening, the external damping force is large and so amplitude decreases significantly. The time period of the oscillations actually increases slightly. An example of heavy dampening is a pendulum oscillating in water.

What happens when there is Critical Dampening?

When there is critical dampening, the external damping force is so large that the oscillator slowly stops at the equilibrium before one oscillation is complete. An example of critical dampening is a pendulum oscillating in treacle.

What are Free Oscillations?

Free Oscillations are where mechanical systems are displaced from their equilibrium position and then allowed to oscillate without any external forces.

What is the Natural Frequency?

When an object oscillates without any external forces being applied (free oscillation) it oscillates at its natural frequency.

What is a Forced Oscillation?

A forced oscillation is an oscillation in which a periodic driver force is applied to an object, which causes it to oscillate at a frequency known as the driving frequency.

What is Resonance?

Resonance is when the driving frequency of the external force applied to an object is the same as the natural frequency of the object, causing the amplitude to rapidly increase. If there is no dampening, the amplitude will continue to increase until the system fails.

How does Barton's Pendulum work?

1. A number of paper cone pendulums of varying lengths are suspended from a string, along with a heavy brass bob D.

2. This heavy pendulum acts as the driver for the paper cone pendulums and oscillates at its natural frequency, forcing all the other pendulums to oscillate at the same frequency.

3. As one of the paper cone pendulums has the same length as pendulum D, it has the same natural frequency. This means the driving force is equal to its natural frequency and so it will resonate.

4. This means that its amplitude will be greater than the other paper cone pendulums.

How can we find the Constant Ratio Property for the Amplitude of a Damped Oscillating system?

In any exponential decay, the physical quantity (amplitude in this case) decreases by the same factor in equal time intervals. For example, for an amplitude A that decays exponentially and is measured every 4 seconds, then A4/A0 = A8/A4 = A12/A8 = Constant.

What can we Infer from the Graph?

For light dampening, the maximum amplitude occurs at the natural frequency f0 of the forced oscillator. As the amount of dampening increases, the amplitude of vibration at any frequency decreases, the maximum amplitude occurs at a lower frequency than f0, and the peak on the graph becomes flatter and broader.

How can we Observe Damped Oscillations? 1/2 PAG

1. Attach a spring to a clamp stand and attach a 500g mass holder to the spring. It's large so that the time period is longer, meaning the maximum amplitude of each oscillation is easier to measure.

2. Wait until the spring stops moving completely and then place a fiducial marker at the bottom of the mass holder. This represents the centre of oscillations and will make it easier to count how many oscillations the mass-spring system has undergone, reducing uncertainty.

3. Attach a 15cm ruler, on either side of the fiducial marker using rubber bands. These will measure the amplitude of the oscillations.

4. Pull the spring down slightly and let it go so that it's oscillating with a small amplitude in a straight line.

5. As the bottom of the mass holder passes the fiducial marker, start a stopwatch and count the time taken for it to complete 10 full oscillations.

6. Repeat the last two steps making sure to measure the amplitude the spring is pulled to, as this will be the maximum amplitude at the start of the first oscillation.

7. Calculate the mean values of maximum amplitude at each oscillation.

8. Repeat the steps above but use a 400g mass holder and a dampening card wedged between the mass holder and a 100g mass.

How can we Observe Damped Oscillations? 2/2 PAG

9. Divide the collected values of the time period by 10 to get the time period for a single oscillation T.

10. Calculate the frequency of the oscillations using f = 1/T when the dampening card is present and when it is not. The frequency with the damping card will be less than the system without the damping card even though mass and spring constant is kept constant. This is because the system experiences more dampening so will move slightly slower.

11. Plot a graph of the maximum amplitude against the number of oscillations for both systems and draw a line of best fit. When comparing the graphs, you should see that the maximum amplitude decreases in both systems, however, the amplitude will decay much faster in the damped system as the degree of damping is greater.

12. Be careful with masses as dropping them can cause injury, wear eye protection when using springs, and add a counterweight to the base of the clamp stand to prevent it from falling.

13. To reduce uncertainty and to more accurately measure the maximum amplitude, you can record the oscillations of both systems using a position sensor connected to a computer.

How can we Observe Forced Oscillations and Resonance? 1/2 PAG

1. Suspend a mass holder on a spring attached to a vibration generator connected to a signal generator using a clamp stand. Beneath the mass holder, place a position sensor connected to a computer.

2. Turn on the signal generator and set it to a frequency much lower than the natural frequency of the spring. (If not known, set it to 10Hz and let spring oscillate freely to calculate natural frequency).

3. Wait until the spring stops moving, then measure the distance of the bottom of the mass holder above the sensor using the position sensor.

4. Using the position sensor connected to a computer with data-logging software, record the maximum amplitude of the oscillations above its equilibrium position.

5. Increase the frequency of the signal generator by 10Hz, and repeat the step above.

6. Repeat the last step until the frequency of the signal generator is far above the frequency of the spring if known, if not, aim for 10 readings.

How can we Observe Forced Oscillations and Resonance? 2/2 PAG

7. Plot a graph of maximum amplitude against frequency. From the graph, you can see that the maximum amplitude of oscillations increases greatly as the frequency of the driving force (vibration generator) comes closer to the natural frequency. This is because resonance occurs when the frequency of the driving force is equal to the natural frequency of the spring.

8. As the driving frequency is equal to the natural frequency of the spring when its experiencing resonance, you can calculate the natural frequency of the spring by finding the frequency at which the maximum amplitude of oscillations reaches its peak value.

9. Be careful with masses as dropping them can cause injury, wear eye protection when using springs, and add a counterweight to the base of the clamp stand to prevent it from falling.