Chapter 6 Oscillations

Question 1

An oscillation is defined as:

Answer 1

Repeated back and forth movements on either side of any equilibrium position

• When the object stops oscillating, it returns to its equilibrium position
  
  • An oscillation is a more specific term for a vibration
  • An oscillator is a device that works on the principles of oscillations
• Oscillating systems can be represented by displacement-time graphs similar to transverse waves
• The shape of the graph is a sine curve
  
  • The motion is described as sinusoidal

Question 2

Properties of Oscillations

Answer 2

• Displacement (x) of an oscillating system is defined as:

The distance of an oscillator from its equilibrium position

• Amplitude (x₀) is defined as:

The maximum displacement of an oscillator from its equilibrium position

• Angular frequency (⍵) is defined as:

The rate of change of angular displacement with respect to time

• This is a scalar quantity measured in rad s^-1 and is defined by the equation:

⍵=2(pie)/T = 2(pie)f

• Frequency (f) is defined as:

The number of complete oscillations per unit time

• It is measured in Hertz (Hz) and is defined by the equation:

f=1/T

• Time period (T) is defined as:

The time taken for one complete oscillation, in seconds

• One complete oscillation is defined as:

The time taken for the oscillator to pass the equilibrium from one side and back again fully from the other side

• The time period is defined by the equation:

T= 1/f = 2(pie)/⍵

Question 3

Phase difference

Answer 3

is how much one oscillator is in front or behind another

• When the relative position of two oscillators are equal, they are in phase
• When one oscillator is exactly half a cycle behind another, they are said to be in anti-phase
• Phase difference is normally measured in radians or fractions of a cycle
• When two oscillators are in antiphase they have a phase difference of π radians

Question 4

Conditions for Simple Harmonic Motion

Answer 4

• it must satisfy the following conditions:
  
  • Periodic oscillations
  • Acceleration proportional to its displacement
  • Acceleration in the opposite direction to its displacement

Question 5

• Simple harmonic motion (SHM) is

Answer 5

• is a specific type of oscillation
• SHM is defined as:

A type of oscillation in which the acceleration of a body is proportional to its displacement, but acts in the opposite direction

Question 6

• Acceleration a and displacement x can be represented by the defining equation of SHM:

Answer 6

a ∝ −x

• An object in SHM will also have a restoring force to return it to its equilibrium position
• This restoring force will be directly proportional, but in the opposite direction, to the displacement of the object from the equilibrium position
• Note: the restoring force and acceleration act in the same direction

Question 7

Calculating Acceleration & Displacement of an Oscillator

Answer 7

• The acceleration of an object oscillating in simple harmonic motion is:

a = −⍵²x

• Where:
  
  • a = acceleration (m s^-2)
  • ⍵ = angular frequency (rad s^-1)
  • x = displacement (m)
• This is used to find the acceleration of an object in SHM with a particular angular frequency ⍵ at a specific displacement x

Question 8

• a = −⍵²x demonstrates:

Answer 8

• The acceleration reaches its maximum value when the displacement is at a maximum ie. x = x₀ (amplitude)
• The minus sign shows that when the object is displacement to the right, the direction of the acceleration is to the left

Question 9

The acceleration of an object in SHM is directly proportional to the negative displacement

Answer 9

• The graph of acceleration against displacement is a straight line through the origin sloping downwards (similar to y = − x)
• Key features of the graph:
  
  • The gradient is equal to − ⍵²
  • The maximum and minimum displacement x values are the amplitudes −x₀ and +x₀
• A solution to the SHM acceleration equation is the displacement equation:

x = x₀sin(⍵t)

• Where:
  
  • x = displacement (m)
  • x₀ = amplitude (m)
  • t = time (s)
• This equation can be used to find the position of an object in SHM with a particular angular frequency and amplitude at a moment in time
• Note: This version of the equation is only relevant when an object begins oscillating from the equilibrium position (x = 0 at t = 0)

Question 10

• The displacement will be at its maximum when sin(⍵t) equals 1 or − 1, when x = x₀
• If an object is oscillating from its amplitude position (x = x₀ or x = − x₀ at t = 0) then the displacement equation will be:

Answer 10

x = x₀cos(⍵t)

• This is because the cosine graph starts at a maximum, whilst the sine graph starts at 0

Question 11

These two graphs represent the same SHM. The difference is the starting position

Answer 11

Question 12

Calculating Speed of an Oscillator

Answer 12

• The greatest speed of an oscillator is at the equilibrium position ie. when its displacement is 0 (x = 0)
• The speed of an oscillator in SHM is defined by:

v = v₀ cos(⍵t)

• Where:
  
  • v = speed (m s^-1)
  • v₀ = maximum speed (m s^-1)
  • ⍵ = angular frequency (rad s^-1)
  • t = time (s)
• This is a cosine function if the object starts oscillating from the equilibrium position (x = 0 when t = 0)
• Although the symbol v is commonly used to represent velocity, not speed, exam questions focus more on the magnitude of the velocity than its direction in SHM

Question 13

• How the speed v changes with the oscillator’s displacement x is defined by:

Answer 13

*equation at bottom

• Where:
  
  • x = displacement (m)
  • x₀ = amplitude (m)
  • ± = ‘plus or minus’. The value can be negative or positive
• This equation shows that when an oscillator has a greater amplitude x₀, it has to travel a greater distance in the same time and hence has greater speed v
• Both equations for speed will be given on your formulae sheet in the exam
• When the speed is at its maximums (at x = 0), the equation becomes:

v₀ = ⍵x₀

Question 14

The variation of the speed of a mass on a spring in SHM over one complete cycle

Answer 14

Question 15

SHM Graphs

Answer 15

• The displacement, velocity and acceleration of an object in simple harmonic motion can be represented by graphs against time
• All undamped SHM graphs are represented by periodic functions
  
  • This means they can all be described by sine and cosine curves

Question 16

Key features of the displacement-time graph (SHM):

Answer 16

• The amplitude of oscillations x₀ can be found from the maximum value of x
• The time period of oscillations T can be found from reading the time taken for one full cycle
• The graph might not always start at 0
• If the oscillations starts at the positive or negative amplitude, the displacement will be at its maximum

Question 17

Key features of the velocity-time graph(SHM)

Answer 17

• • It is 90^o out of phase with the displacement-time graph
    
    • Velocity is equal to the rate of change of displacement
    • So, the velocity of an oscillator at any time can be determined from the gradient of the displacement-time graph:

v= ∇x/∇t

• An oscillator moves the fastest at its equilibrium position
• Therefore, the velocity is at its maximum when the displacement is zero

Question 18

Key features of the acceleration-time graph(SHM):

Answer 18

• • The acceleration graph is a reflection of the displacement graph on the x axis
    
    • This means when a mass has positive displacement (to the right) the acceleration is in the opposite direction (to the left) and vice versa
    • It is 90^o out of phase with the velocity-time graph
    • Acceleration is equal to the rate of change of velocity
    • So, the acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph:

a= ∇v/∇t

• • The maximum value of the acceleration is when the oscillator is at its maximum displacement

Question 19

The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other

Answer 19

Question 20

Kinetic & Potential Energies

Answer 20

Speed v is at a maximum when displacement x = 0, so:

The kinetic energy is at a maximum when the displacement x = 0 (equilibrium position)

• Therefore, the kinetic energy is 0 at maximum displacement x = x₀, so:

The potential energy is at a maximum when the displacement (both positive and negative) is at a maximum x = x₀ (amplitude)

• A simple harmonic system is therefore constantly converting between kinetic and potential energy
  
  • When one increases, the other decreases and vice versa, therefore:

The total energy of a simple harmonic system always remains constant and is equal to the sum of the kinetic and potential energies

Question 21

• During simple harmonic motion, energy is constantly exchanged between two forms: kinetic and potential

Answer 21

• The potential energy could be in the form of:
  
  • Gravitational potential energy (for a pendulum)
  • Elastic potential energy (for a horizontal mass on a spring)
  • Or both (for a vertical mass on a spring)

Question 22

The kinetic and potential energy of an oscillator in SHM vary periodically

Answer 22

Question 23

The key features of the energy-time graph:

Answer 23

Both the kinetic and potential energies are represented by periodic functions (sine or cosine) which are varying in opposite directions to one another

• When the potential energy is 0, the kinetic energy is at its maximum point and vice versa
• The total energy is represented by a horizontal straight line directly above the curves at the maximum value of both the kinetic or potential energy
• Energy is always positive so there are no negative values on the y axis
• Note: kinetic and potential energy go through two complete cycles during one period of oscillation
• This is because one complete oscillation reaches the maximum displacement twice (positive and negative)

Question 24

The key features of the energy-displacement graph:

Answer 24

• Displacement is a vector, so, the graph has both positive and negative x values
• The potential energy is always at a maximum at the amplitude positions x₀ and 0 at the equilibrium position (x = 0)
• This is represented by a ‘U’ shaped curve
• The kinetic energy is the opposite: it is 0 at the amplitude positions x₀ and maximum at the equilibrium position x = 0
• This is represented by a ‘n’ shaped curve
• The total energy is represented by a horizontal straight line above the curves

Question 25

***Graph showing the potential and kinetic energy against displacement in half a period of an SHM oscillation***

Answer 25

Question 26

**Calculating Total Energy of a Simple Harmonic System**

Answer 26

* The total energy of system undergoing **simple harmonic motion** is defined by: **E = ½ m⍵²x_0²** * Where: * E = total energy of a simple harmonic system (J) * m = mass of the oscillator (kg) * ⍵ = angular frequency (rad s^-1) * x₀ = amplitude (m)

Question 27

**Damping**

Answer 27

***The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system*** * Damping continues until the oscillator comes to rest at the equilibrium position * A key feature of simple harmonic motion is that the **frequency** of damped oscillations **does not change** as the amplitude decreases

Question 28

In practice, all oscillators eventually stop oscillating

Answer 28

* * Their amplitudes decrease rapidly, or gradually * This happens due to **resistive forces**, such friction or air resistance, which act in the opposite direction to the motion of an oscillator * Resistive forces acting on an oscillating simple harmonic system cause **damping** * These are known as **damped** oscillations

Question 29

***Damping on a mass on a spring is caused by a resistive force acting in the opposite direction to the motion. This continues until the amplitude of the oscillations reaches zero***

Answer 29

Question 30

make sure not to confuse resistive force and restoring force

Answer 30

**-resistive** force is what opposes the motion of the oscillator and causes damping **-restoring** force is what brings the oscillator back to the equilibrium position

Question 31

**Types of Damping**

Answer 31

* There are three degrees of damping depending on how quickly the amplitude of the oscillations decrease: * Light damping * Critical damping * Heavy damping

Question 32

Light Damping

Answer 32

* When oscillations are lightly damped, the amplitude does not decrease linearly * It **decays exponentially** with time * When a lightly damped oscillator is displaced from the equilibrium, it will oscillate with gradually decreasing amplitude

Question 33

**Key features of a displacement-time graph for a lightly damped system:**

Answer 33

* There are many oscillations represented by a sine or cosine curve with gradually decreasing amplitude over time * This is shown by the height of the curve decreasing in both the positive and negative displacement values * The amplitude decreases exponentially * The frequency of the oscillations remain constant, this means the time period of oscillations must stay the same and each peak and trough is equally spaced

Question 34

***A graph for a lightly damped system consists of oscillations decreasing exponentially***

Answer 34

Question 35

Critical Damping

Answer 35

* When a critically damped oscillator is displaced from the equilibrium, it will return to rest at its equilibrium position in the shortest possible time **without** oscillating * For example, car suspension systems prevent the car from oscillating after travelling over a bump in the road

Question 36

**Key features of a displacement-time graph for a critically damped system:**

Answer 36

* This system does **not** oscillate, meaning the displacement falls to 0 straight away * The graph has a fast decreasing gradient when the oscillator is first displaced until it reaches the x axis * When the oscillator reaches the equilibrium position (x = 0), the graph is a horizontal line at x = 0 for the remaining time

Question 37

***The graph for a critically damped system shows no oscillations and the displacement returns to zero in the quickest possible time***

Answer 37

Question 38

Heavy Damping

Answer 38

* When a heavily damped oscillator is displaced from the equilibrium, it will take a long time to return to its equilibrium position **without** oscillating * The system returns to equilibrium more slowly than the critical damping case * For example, door dampers to prevent them slamming shut

Question 39

**Key features of a displacement-time graph for a heavily damped system:**

Answer 39

* There are no oscillations. This means the displacement does not pass 0 * The graph has a slow decreasing gradient from when the oscillator is first displaced until it reaches the x axis * The oscillator reaches the equilibrium position (x = 0) after a long period of time, after which the graph remains a horizontal line for the remaining time

Question 40

***A heavy damping curve has no oscillations and the displacement returns to zero after a long period of time***

Answer 40

Question 41

**Resonance**

Answer 41

***When the driving frequency applied to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations increases significantly*** * For example, when a child is pushed on a swing: * The swing plus the child has a fixed natural frequency * A small push after each cycle increases the amplitude of the oscillations to swing the child higher * If the driving frequency does not quite match the natural frequency, the amplitude will increase but not to the same extent at when resonance is achieved * This is because at resonance, energy is transferred from the driver to the oscillating system **most efficiently** * Therefore, at resonance, the system will be transferring the maximum kinetic energy possible

Question 42

* These are known as **forced oscillations**, and are defined as:

Answer 42

***Periodic forces which are applied in order to sustain oscillations*** * For example, when a child is on a swing, they will be pushed at one end after each cycle in order to keep swinging and prevent air resistance from damping the oscillations * These extra pushes are the forced oscillations, without them, the child will eventually come to a stop * The frequency of forced oscillations is referred to as the **driving frequency** **(f)**

Question 43

* All oscillating systems have a **natural frequency (f₀)**, this is defined as:

Answer 43

***The frequency of an oscillation when the oscillating system is allowed to oscillate freely*** * Oscillating systems can exhibit a property known as **resonance** * When resonance is achieved, a maximum amplitude of oscillations can be observed

Question 44

A graph of driving frequency *f* against amplitude *a* of oscillations is called a **resonance curve**. It has the following key features:

Answer 44

* When f \< f₀, the amplitude of oscillations increases * At the peak where f = f₀, the amplitude is at its maximum. This is **resonance** * When f \> f₀, the amplitude of oscillations starts to decrease

Question 45

***The maximum amplitude of the oscillations occurs when the driving frequency is equal to the natural frequency of the oscillator***

Answer 45

Question 46

Damping reduces the??

Answer 46

* amplitude of resonance vibrations * The height and shape of the resonance curve will therefore change slightly depending on the degree of damping * **Note:** the natural frequency *f₀* will remain the same

Question 47

As the degree of damping is increased, the resonance graph is altered in the following ways:

Answer 47

* The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers * The resonance peak broadens * The resonance peak moves slightly to the left of the natural frequency when heavily damped

Question 48

***As damping is increased, resonance peak lowers, the curve broadens and moves slightly to the left***

Answer 48