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Physics A Level CIE > Chapter 6 Oscillations > Flashcards

Chapter 6 Oscillations Flashcards

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1
Q

An oscillation is defined as:

A

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
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2
Q

Properties of Oscillations

A
  • 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)/⍵

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3
Q

Phase difference

A

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
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4
Q

Conditions for Simple Harmonic Motion

A
  • it must satisfy the following conditions:
    • Periodic oscillations
    • Acceleration proportional to its displacement
    • Acceleration in the opposite direction to its displacement
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5
Q
  • Simple harmonic motion (SHM) is
A
  • 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

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6
Q
  • Acceleration a and displacement x can be represented by the defining equation of SHM:
A

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
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7
Q

Calculating Acceleration & Displacement of an Oscillator

A
  • 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
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8
Q
  • a = −⍵2x demonstrates:
A
  • 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
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9
Q

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

A
  • 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)
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10
Q
  • 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:
A

x = x0cos(⍵t)

  • This is because the cosine graph starts at a maximum, whilst the sine graph starts at 0
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11
Q

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

A
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12
Q

Calculating Speed of an Oscillator

A
  • 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
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13
Q
  • How the speed v changes with the oscillator’s displacement x is defined by:
A

*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

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14
Q

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

A
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15
Q

SHM Graphs

A
  • 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
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16
Q

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

A
  • 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
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17
Q

Key features of the velocity-time graph(SHM)

A
    • It is 90o 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
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18
Q

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

A
    • 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 90o 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
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19
Q

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

A
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20
Q

Kinetic & Potential Energies

A

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

21
Q
  • During simple harmonic motion, energy is constantly exchanged between two forms: kinetic and potential
A
  • 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)
22
Q

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

A
23
Q

The key features of the energy-time graph:

A

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)
24
Q

The key features of the energy-displacement graph:

A
  • 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
25
Q

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

A
26
Q

Calculating Total Energy of a Simple Harmonic System

A
  • The total energy of system undergoing simple harmonic motion is defined by:

E = ½ m⍵2x02

  • Where:
    • E = total energy of a simple harmonic system (J)
    • m = mass of the oscillator (kg)
    • ⍵ = angular frequency (rad s-1)
    • x0 = amplitude (m)
27
Q

Damping

A

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
28
Q

In practice, all oscillators eventually stop oscillating

A
    • 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
29
Q

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

A
30
Q

make sure not to confuse resistive force and restoring force

A

-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

31
Q

Types of Damping

A
  • There are three degrees of damping depending on how quickly the amplitude of the oscillations decrease:
    • Light damping
    • Critical damping
    • Heavy damping
32
Q

Light Damping

A
  • 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
33
Q

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

A
  • 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
34
Q

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

A
35
Q

Critical Damping

A
  • 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
36
Q

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

A
  • 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
37
Q

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

A
38
Q

Heavy Damping

A
  • 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
39
Q

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

A
  • 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
40
Q

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

A
41
Q

Resonance

A

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
42
Q
  • These are known as forced oscillations, and are defined as:
A

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)
43
Q
  • All oscillating systems have a natural frequency (f0), this is defined as:
A

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
44
Q

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

A
  • 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
45
Q

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

A
46
Q

Damping reduces the??

A
  • 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
47
Q

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

A
  • 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
48
Q

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

A

Physics A Level CIE (14 decks)

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