Pearson Edexcel Physics

Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology

5.5 Oscillations

Pearson Edexcel Physics

Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology

5.5 Oscillations

Candidates will be assessed on their ability to::
143. Understand that the condition for simple harmonic motion is F = − kx, and hence understand how to identify situations in which simple harmonic motion will occur
144.

Be able to use the equations

[math]\alpha=-\omega^2 x,x=-A \cos \omega t,x=-A\omega \sin \omega t,a=-A\omega^2 \cos \omega t,\text{ and } T=\frac{1}{f}=\frac{2\pi}{\omega} \text{ and } \omega=2\pi f[/math]

As applied to a simple harmonic oscillator
145.

Be able to use equations for a simple harmonic oscillator

[math]T = 2\pi \sqrt{\frac{m}{k}}[/math]

And a simple pendulum

[math]T = 2\pi \sqrt{\frac{l}{g}}[/math]
146. Be able to draw and interpret a displacement-time graph for an object oscillating and know that the gradient at a point gives the velocity at that point
147. Be able to draw and interpret a velocity-time graph for an oscillating object and know that the gradient at a point gives the acceleration at that point
148. Understand what is meant by resonance
149. CORE PRACTICAL 16: Determine the value of an unknown mass using the resonant frequencies of the oscillation of known masses
150. Understand how to apply conservation of energy to damped and undamped oscillating systems
151. Understand the distinction between free and forced oscillations
152. Understand how the amplitude of a forced oscillation changes at and around the natural frequency of a system and know, qualitatively, how damping affects resonance
153. Understand how damping and the plastic deformation of ductile materials reduce the amplitude of oscillation.

  • 143) Understand that the condition for simple harmonic motion is F = -kx and hence understand how to identify situations in which simple harmonic motion will occur

  • ⇒ Bouncing and swinging:
  • There are many things around us which oscillate (figure 1). This means they have continuously repeated movements. For example, a child’s swing goes backwards and forwards through the same positions over and over again.


    Figure 1 How quickly does a bouncing spring toy move?

  • If it swings freely, it will always take the same time to complete one full swing. This is known as its time period, 7. It follows a system of movements known as simple harmonic motion (SHM).
  • When a system is moving in SHM, a force, known as a restoring force, F, is trying to return the object to its equilibrium position, and this force is proportional to the displacement, x, from that equilibrium position.
  • [math]F = -kx[/math]
  • Where k is a constant dependent on the oscillating system in question.
  • The negative sign in the equation shows us that the acceleration will always be towards the Centre of oscillation.
  • The SHM definition is the same equation as in Hooke’s law, and the oscillation of a mass attached to a spring is an example of SHM.

  • 144) Be able to use the equations

  • [math]\alpha=-\omega^2 x,x=-A \cos \omega t,x=-A\omega \sin \omega t,a=-A\omega^2 \cos \omega t,\text{ and } T=\frac{1}{f}=\frac{2\pi}{\omega} \text{ and } \omega=2\pi f[/math]

  •  As applied to a simple harmonic oscillator

  • ⇒ Angular velocity and SHM:
  • For objects moving in a circle, the relationship that gave us the angular velocity is:
  • [math]\omega = \frac{\theta}{t}[/math]
  • Figure 2 Relationships in circular motion.
  • The relationships between time and frequency:
  • [math]T = \frac{1}{f} \\
    T = \frac{2\pi}{\omega}[/math]
  • Therefore, for an object moving in a circle:
  • [math]ω = 2πf[/math]
  • The movement of an oscillating object follows a similar pattern to that of circular motion, so all of these equations are valid in SHM.
  • For an object performing SHM, we can determine its angular velocity, despite the fact that it may not actually be moving in a circle. Its motion is the projection of motion in a circle.
  • In fig A, consider the horizontally projected distance to the object from the vertical axis, and how these changes over time as the object rotates around the circle.
  • When the object is at A, this projected distance is equal to the radius of the circle, r, but at position B this distance is shown by OC. You could calculate OC from:
  • [math]x = r cos⁡θ[/math]
  • As the object moves around the circle, its angular displacement changes according to its angular velocity,
  • [math]\omega = \frac{\theta}{t}[/math]
  • Rearranging this as
  • [math]θ = ωt[/math]
  • The displacement can be rewritten as:
  • [math]x = r cos⁡ωt[/math]
  • The motion of all simple harmonic oscillators can be described by an equation of this form. Indeed, all simple harmonic oscillators can be described by a sine or cosine function which gives their displacement, velocity and acceleration over time.
  • In the case of a pendulum bob, the radius of the circle is replaced by the amplitude of the pendulum’s swing. The expression for the displacement becomes:
  • [math]x = r cos⁡ωt \qquad \text{because at} t = 0, x = A[/math]
  • ⇒ SHM Graphs
  • When an object is moving with SHM the restoring force is proportional to the displacement,
  • [math]F = – kx[/math]
  • As we can describe the position using
  • [math]x = A cos ωt[/math]
  • this gives the equation for the force overtime as:
  • [math]F = -kA cos ωt[/math]
  • Not all SHM oscillations involve a spring, so here k refers to some constant relevant to the oscillator set-up. From Newton’s second law, we can also show that:
  • [math]\begin{gather}
    ma = -kx \\
    a = \frac{-kx}{m} \\
    a = -\frac{k}{m} (A \cos \omega t)
    \end{gather}[/math]
  • From this we can see that the acceleration and displacement in SHM have the same form, but the acceleration acts in the opposite direction to the displacement.
  • When the displacement is zero, so is the acceleration. And when x is at its maximum value, the acceleration is also at its maximum value figure 3
  • The velocity of an oscillator at any moment from the gradient of the displacement-time graph. For a function, [math]x = sin⁡θ[/math]
  • The derivative function, [math]\frac{dx}{dt}[/math],which tells us the gradient at each point,
  • [math]\frac{dx}{dt} = -\sin \theta \quad \text{(Remember, } \theta \text{ is a function of time)}[/math]
  • If [math]x = A cos ωt[/math], then the change in displacement with time is given by:
  • [math]\begin{gather}
    \frac{dx}{dt} = v = -A\omega \sin \omega t
    \end{gather}[/math]
  • The change in velocity with time, the acceleration is:
  • [math]\begin{gather}
    \frac{d^2 x}{dt^2} + \frac{dv}{dt} = a = -A\omega^2 \cos \omega t
    \end{gather}[/math]
  • Or
  • [math]a = -ω^2 x[/math]
  • If we know the mass of the oscillating object, and the angular velocity of the oscillation, we can work out the restoring force constant, k, in any SHМ.
  • [math]\begin{gather}
    a = -\frac{kx}{m} = -\omega^2 x \\
    k = \omega^2 m
    \end{gather}[/math]
  • Figure 3 The changes in position, velocity and acceleration over time for a simple harmonic oscillator. The displacement can be described using either sine or cosine (depending upon whether the oscillations are taken to start from the Centre or at amplitude). Velocity and acceleration are then also sine or cosine as appropriate.

  • 145) Be able to use equations for a simple harmonic oscillator

  • [math]T = 2\pi \sqrt{\frac{m}{k}}[/math]

  • And a simple pendulum

  • [math]T = 2\pi \sqrt{\frac{l}{g}}[/math]

  • The SHM definition is the same equation as in Hooke’s law, and the oscillation of a mass attached to a spring is an example of SHM.
  • The equation for the period of the oscillations of a mass, m, subject to a Hooke’s law restoring force:
  • [math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
  • ⇒ Pendulum Dynamics:
  • Apparently, when Galileo was just 17, he was watching a hanging lantern in the cathedral in Pisa and he observed that a pendulum’s time period is independent of the size of the oscillations.
  • In fact, the period for a pendulum is given by the expression:
  • [math]T = 2\pi \sqrt{\frac{l}{g}}[/math]
  • Therefore, the period is only dependent on the length, 1, of the pendulum string, and the strength of gravity on the planet on which it has been set up.
  • Example:
  • a) Figure 4 A 500 g toy train is attached to a pole by a spring, with a spring constant of [math]100 N m^{-1}[/math], and made to oscillate horizontally.
  • What force will act on the train when it is at its amplitude position of 8 cm from equilibrium?
  • [math]\begin{gather}
    F = -kx \\
    F = -(100)(0.08) \\
    F = -8\, \text{N}
    \end{gather}[/math]
  • A force of 8 newtons will act on the train, trying to pull it back towards the equilibrium position.
  • b) How fast will the train accelerate whilst at this amplitude position?
  • From Newton’s second law:
  • [math]\begin{gather}
    a = \frac{F}{m} \\
    a = \frac{-8}{0.5} \\
    a = -16 \,\text{m/s}^2
    \end{gather}[/math]
  • c) What is the period of SHM oscillations for the train?
  • [math]\begin{gather}
    T = 2\pi \sqrt{\frac{m}{k}} \\
    T = 2\pi \sqrt{\frac{0.5}{100}} \\
    T = 0.44 \,\text{s}
    \end{gather}[/math]
  • Figure 4 The restoring force in a mass-spring system.

  • 146) Be able to draw and interpret a displacement-time graph for an object oscillating and know that the gradient at a point gives the velocity at that point

  • ⇒ Displacement-Time Graph for Oscillations
  • Oscillation:
  • An oscillation is a repetitive back-and-forth motion about an equilibrium position (like a pendulum, a mass on a spring, or a vibrating tuning fork).
  • The object moves through maximum positive and negative displacement (amplitude) over time.
  • The motion is often sinusoidal (like a sine or cosine wave).
  • Displacement-Time Graph:
  • The graph shows how the displacement (x) of the object varies with time (t).
    1. Wave shape: Sine or cosine curve
    2. Amplitude: Maximum displacement from equilibrium (peak value)
    3. Period (T): Time taken for one complete oscillation (peak to peak)
    4. Frequency (f): Number of oscillations per second
  • [math]f = \frac{1}{T}[/math]
  • A Displacement-Time Graph
  • Figure 5 Displacement-time graph
  • – The object starts at maximum positive displacement, moves to zero, then maximum negative, and back.
  • – This is simple harmonic motion (SHM).
  • The gradient (slope) of a displacement-time graph gives the velocity at any point.
  • Mathematically:
  • [math]v = \frac{dx}{dt}[/math]
  • So:
  • – A steep slope means high speed.
  • – A flat slope (horizontal) means zero velocity.
  • – A positive slope means the object is moving in the positive direction.
  • – A negative slope means it’s moving in the negative direction.

  • 147) Be able to draw and interpret a velocity-time graph for an oscillating object and know that the gradient at a point gives the acceleration at that point

  • ⇒  Velocity-Time Graph for Oscillations
  • The velocity-time graph shows how the velocity of an oscillating object changes with time.
  • In SHM, the motion is sinusoidal—so the velocity also varies sinusoidally, just like displacement, but it is out of phase.
  • ⇒   Shape of the Velocity-Time Graph
  • If the displacement-time graph is a sine wave, the velocity-time graph is a cosine wave (and vice versa), but shifted by a quarter of a period (T/4).
  • A velocity-time graph:
  • Figure 6 Velocity-time graph
  • – The curve is sinusoidal.
  • – Velocity is zero when displacement is at maximum.
  • – Velocity is maximum (positive or negative) when displacement is zero.
  • Gradient = Acceleration
  • The gradient of a velocity-time graph gives the acceleration at that point.
  • [math]\bar{a} = \frac{d\bar{v}}{dt}[/math]
  • So:
Slope Type Acceleration
Positive slope Positive acceleration
Negative slope Negative acceleration
Zero slope Zero acceleration
  • ⇒  Interpreting the Velocity-Time Graph
Point on Graph Velocity Gradient (Acceleration) Meaning
Crosses zero 0 Max (±) Acceleration is greatest; object turning
Peaks (top/bottom) ±V 0 Acceleration = 0; object passing through equilibrium
  • ⇒  Relationship with Other Graphs
Graph Type Quantity Shown Related Quantity (Gradient)
Displacement–time Displacement Velocity
Velocity–time Velocity Acceleration
Acceleration–time Acceleration –
  • In SHM, these are all sine/cosine functions with phase differences:
  • – Displacement leads to velocity, which leads to acceleration.
  • The maximum values of sin and cos are both 1. So, the maximum values of velocity and acceleration are:
  • [math]\begin{gather}
    v_{\max} = A \omega_{\max} \\
    a_{\max} = -A \omega^2
    \end{gather}[/math]
  • Example:
  • A science museum has a giant demonstration tuning fork. The end of each prong vibrates in simple harmonic motion with a time period of 1.20 s and starts vibrating from an amplitude of 80 cm. Calculate the displacement, velocity and acceleration of a prong after 5 s.
  • A = 0.8 m ;   T = 1.20 s
  • [math]\begin{gather}
    \omega = \frac{2\pi}{T} \\
    \omega = \frac{2(3.14)}{1.20} \\
    \omega = 5.24 \,\text{rad/s} \\
    x = A \cos \omega t \\
    x = (0.80) \cos (5.24)(5) \\
    x = 0.386 \,\text{m} \\
    v = -A \omega \sin \omega t \\
    v = -(0.8)(5.24) \sin (5.24)(5) \\
    v = -3.67 \,\text{m/s} \\
    a = -A \omega^2 \cos \omega t \\
    a = -\omega^2 x \\
    a = -(5.24)^2 (0.386) \\
    a = -10.6 \,\text{m/s}^2
    \end{gather}[/math]

  • 148) Understand what is meant by resonance

  • If a system or object is forced to vibrate at its natural frequency, it will absorb more and more energy, resulting in very large amplitude oscillations. This is resonance.
  • If your washing machine is very noisy during one part of the wash cycle, the motor will be spinning at a certain frequency.
  • This drives vibrations in the machine at a frequency that is the natural frequency of one of the panels, which then vibrates with very large amplitude, making a loud noise.
  • During the rest of the wash, the motor’s rotation generates vibrations at other frequencies that do not match with the natural frequency of any part of the machine, and it is much quieter.
  • Figure 7 Graphical illustration of how the driving frequency in a washing machine affects the oscillation amplitude of its side panel. At all frequencies here, the amplitude of forced vibration is the same, and relatively small. When the motor’s vibration reaches about 1100 Hz, the graph peaks, showing that the side panel vibrates significantly, which will be very noisy.

  • 149) CORE PRACTICAL 16: Determine the value of an unknown mass using the resonant frequencies of the oscillation of known masses

  • The set-up in figure 8 allows us to monitor the amplitude of vibrations of a mass on a spring, as we force it to vibrate at different frequencies.
  • By monitoring for a very large amplitude of vibration, we can determine the resonant (or natural) frequency for the set-up.
  • Figure 8 Forced oscillations can be set up at a desired driving frequency.
  • From previously, we know that a mass oscillating on a spring follows the equation for its time period:
  • [math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
  • The time period will be the reciprocal of the natural frequency, f, and this is also the resonant frequency:
  • [math]f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}[/math]
  • Thus, by finding the resonant frequency, from monitoring the amplitude of forced oscillations, we could determine an unknown mass that is hanging on the spring:
  • [math]m = \frac{k}{4 \pi^2 f^2}[/math]
  • The spring constant, k, could be found by measuring the resonant frequency for a known mass and rearranging the equation to find k:
  • [math]k = 4π^2 f^2 m[/math]

  • 150) Understand how to apply conservation of energy to damped and undamped oscillating systems

  • ⇒ Conservation of Energy:
  • An oscillator is a closed system which means that it cannot gain or lose energy, unless there is an external influence.
  • For example, a pendulum under water will not continue to swing for very long.
  • This is because the viscous drag on the pendulum from the water provides an external force, which can do work to remove energy from the pendulum system.
  • A child on a swing can continue to oscillate, despite air resistance removing energy, if there is also a parent pushing repeatedly (doing work) to put energy back into the system.
  • Such damped or forced oscillations will be dealt with in the next sections. Here we will only consider a closed system in which no energy moves in or out of the system during its oscillations.
  • ⇒ Energy transfers during oscillation:
  • The kinetic energy of a pendulum varies during its swinging motion.
  • At each end of the swing, the bob is stationary for an instant, meaning that it has zero kinetic energy at these points.
  • The kinetic energy then increases steadily until it reaches a maximum, when the bob passes through the central position and is moving with its maximum speed.
  • The kinetic energy then decreases to reach zero at the other amplitude position. This cycle repeats continuously.
  • However, once swinging, the pendulum system cannot gain or lose energy. The varying kinetic energy must be transferring back and forth into another store.
  • In this example, the other store is gravitational potential energy as the pendulum rises to a maximum height, where it has maximum GPE at each end of the swing and drops to minimum GPE as it passes through the lowest point (the central position). These energy changes are shown in figure 9.
  • Figure 9 The energy in a simple harmonic oscillator is constantly being transferred from kinetic to potential and back again, whilst the total remains constant as the sum of the kinetic and potential energies at each moment.
  • When stretched a little before release, the spring has stored elastic potential energy.
  • On release, the spring accelerates the mass back towards the equilibrium position, where it has maximum kinetic energy and zero potential.
  • This kinetic energy is then transferred into elastic potential as the spring squashes in the other displacement direction.
  • The transfers of potential and kinetic energies continue, and if no energy can be lost from the system it would oscillate with the same amplitude forever.
  • The above discussions are exclusively for a system that cannot lose any energy. The closest we can get to this system would be a pendulum, or mass on a spring, in a vacuum.
  • These would be almost free from energy losses, but air resistance will act in most real situations.
  • Example:
  • A forest playground has a tyre hanging from a tree branch. The tyre behaves like a pendulum, with a rope of 4.0 metres length, and its mass is 15 kg.
  • A child of mass 45 kg swings on the tyre by pulling it 3.0 metres to one side (amplitude) and leaping on. What is the maximum height that the tyre reaches above its equilibrium position?
  • [math]\begin{gather}
    T = 2\pi \sqrt{\frac{l}{g}} \\
    T = 2\pi \sqrt{\frac{4}{9.8}} \\
    T = 2(3.14)(0.64) \\
    T = 4 \,\text{s} \\
    \omega = \frac{2\pi}{T} \\
    \omega = \frac{2(3.14)}{4} \\
    \omega = 1.6 \,\text{rad/s} \\
    v = -A \omega \sin \omega t
    \end{gather}[/math]
  • At maximum velocity, sin⁡ωt will equal 1, so:
  • [math]\begin{gather}
    v_{\max} = A \omega \\
    v_{\max} = (3)(1.6) \\
    v_{\max} = 4.8 \,\text{m/s}
    \end{gather}[/math]
  • Maximum kinetic energy:
  • [math]\begin{gather}
    E_{k\max} = \frac{1}{2} m v_{\max}^2 \\
    E_{k\max} = \frac{1}{2} (45 + 15) (4.8)^2 \\
    E_{k\max} = \frac{1}{2} (45 + 15)(23.04) \\
    E_{k\max} = \frac{1}{2} (60)(23.04) \\
    E_{k\max} = 691.2 \,\text{J}
    \end{gather}[/math]

  • 151) Understand the distinction between free and forced oscillations

  • ⇒ Free and forced oscillations:
  • Releasing a pendulum from its maximum amplitude and letting it swing freely (preferably in a vacuum) is an example of free oscillation.
  • The situation is set up for a continuous exchange of potential and kinetic energy, caused by a restoring force which is proportional to the displacement.
  • Any oscillating system has a natural frequency – the frequency at which it naturally chooses to oscillate when left alone.
  • However, oscillators can be forced to behave in a different way to their natural motion. If, as a pendulum swings one way, you push in the opposite direction, it turns back.
  • By repeated applications of forces from your hand in different directions, you could force the pendulum to oscillate at some other frequency.
  • This would then be forced oscillation – not SHM – and the frequency at which you were causing it to swing at would be your driving frequency.
  • Forcing oscillations involves adding energy to a system whilst it oscillates. Unless this is done at the natural frequency, the system is unlikely to undergo SHM and will dissipate the energy quite quickly.
  • This is what happens if you push a child on a swing at the wrong moment in the oscillation.
  • Figure 10 Free and force oscillation

  • 152) Understand how the amplitude of a forced oscillation changes at and around the natural frequency of a system and know, qualitatively, how damping affects resonance

  • ⇒ Damped Oscillations:
  • Damped oscillations suffer a loss in energy in each oscillation and this reduces the amplitude over time (fig C).
  • If a system is performing SHM at its natural frequency, its energy may be dissipated through a friction force acting on the system, or the plastic deformation of a ductile material in the system.
  • If a pendulum is left to swing without interference, its amplitude will constantly decrease with each swing due to air resistance and internal stresses within the flexing material of the string.
  • These effects can be amplified if we attach a small sail to catch the air. This artificial increasing of the air resistance is an example of damping (or in fact an increase in the damping, as there would already be a tiny air resistance force on the pendulum).
  • Note that although the amplitude decreases, the period remains constant throughout. The amount of damping may vary, which will change how quickly the amplitude is reduced. If the oscillator completes several oscillations, the amplitude will decrease exponentially. This is known as underdamping (sometimes called light damping).
  • Swinging the pendulum in a bowl of water will make its amplitude of oscillation drop very rapidly, and it might not even complete one cycle. This is known as overdamping.
  • Figure 11 Damping reduces oscillation amplitude. A bungee jumper’s amplitude decreases because elastic stresses in the rubber rope dissipate energy as heat, and air resistance removes kinetic energy.

  • 153) Understand how damping and the plastic deformation of ductile materials reduce the amplitude of oscillation.

  • Damping is the process by which energy is gradually lost from an oscillating system, usually due to resistive forces like friction or air resistance.
  • Effect on Oscillations:
  • – It causes the amplitude of oscillations to decrease over time.
  • – The system eventually comes to rest if no external energy is added.
  • Damping Reduces Amplitude:
  • – Each cycle, some energy is lost (converted to heat, for example).
  • – Less energy → smaller maximum displacement → reduced amplitude.
  • – The more damping, the faster the amplitude drops.
  • Figure 12 Damping reduces amplitude
  • Example:
  • A swinging pendulum in air will slow down and stop because of air resistance (a damping force).
  • ⇒  Plastic Deformation:
  • Plastic deformation occurs when a ductile material (like copper or soft metal) is permanently stretched or bent after exceeding its elastic limit.
  • – Elastic deformation: Temporary—returns to original shape.
  • – Plastic deformation: Permanent—shape is changed.
  • Effect on Oscillations:
  • When a material undergoes plastic deformation:
  • – It absorbs energy from the oscillation.
  • – This energy is not returned to the system (it’s used to rearrange atoms).
  • – As a result, the amplitude drops.
Factor What It Does Energy Lost As Effect on Amplitude
Damping Opposes motion via resistive forces Heat (friction, drag) Gradual reduction
Plastic Deformation Material stretches permanently under stress Atomic rearrangement Sudden or step-like drop
  • Real-World Example:
  • Car suspension systems use damping to absorb bumps and stop the car from bouncing.
  • Metal wires or springs in machinery may deform plastically under repeated stress, absorbing oscillation energy and reducing motion.
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