Chapter 16 Oscillations(A-level=required)

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

In practice, all oscillators eventually stop oscillating

Answer 27

• • 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 28

make sure not to confuse resistive force and restoring force

Answer 28

-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 29

Resonance

Answer 29

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 30

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

Answer 30

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 31

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

Answer 31

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