Chapter 6 Oscillations Flashcards
An oscillation is defined as:
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
Properties of Oscillations
- 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)/⍵
Phase difference
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
Conditions for Simple Harmonic Motion
- it must satisfy the following conditions:
- Periodic oscillations
- Acceleration proportional to its displacement
- Acceleration in the opposite direction to its displacement
- Simple harmonic motion (SHM) is
- 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
- Acceleration a and displacement x can be represented by the defining equation of SHM:
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
Calculating Acceleration & Displacement of an Oscillator
- 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
- a = −⍵2x demonstrates:
- 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
The acceleration of an object in SHM is directly proportional to the negative displacement
- 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)
- 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:
x = x0cos(⍵t)
- This is because the cosine graph starts at a maximum, whilst the sine graph starts at 0
These two graphs represent the same SHM. The difference is the starting position
Calculating Speed of an Oscillator
- 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
- How the speed v changes with the oscillator’s displacement x is defined by:
*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
The variation of the speed of a mass on a spring in SHM over one complete cycle
SHM Graphs
- 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
Key features of the displacement-time graph (SHM):
- 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
Key features of the velocity-time graph(SHM)
- 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:
- It is 90o out of phase with 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
Key features of the acceleration-time graph(SHM):
- 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:
- The acceleration graph is a reflection of the displacement graph on the x axis
a= ∇v/∇t
- The maximum value of the acceleration is when the oscillator is at its maximum displacement
The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other
Kinetic & Potential Energies
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
- During simple harmonic motion, energy is constantly exchanged between two forms: kinetic and potential
- 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)
The kinetic and potential energy of an oscillator in SHM vary periodically
The key features of the energy-time graph:
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)
The key features of the energy-displacement graph:
- 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
Graph showing the potential and kinetic energy against displacement in half a period of an SHM oscillation
Calculating Total Energy of a Simple Harmonic System
- 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)
Damping
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
In practice, all oscillators eventually stop oscillating
- 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
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
make sure not to confuse resistive force and restoring force
-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
Types of Damping
- There are three degrees of damping depending on how quickly the amplitude of the oscillations decrease:
- Light damping
- Critical damping
- Heavy damping
Light Damping
- 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
Key features of a displacement-time graph for a lightly damped system:
- 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
A graph for a lightly damped system consists of oscillations decreasing exponentially
Critical Damping
- 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
Key features of a displacement-time graph for a critically damped system:
- 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
The graph for a critically damped system shows no oscillations and the displacement returns to zero in the quickest possible time
Heavy Damping
- 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
Key features of a displacement-time graph for a heavily damped system:
- 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
A heavy damping curve has no oscillations and the displacement returns to zero after a long period of time
Resonance
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
- These are known as forced oscillations, and are defined as:
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)
- All oscillating systems have a natural frequency (f0), this is defined as:
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
A graph of driving frequency f against amplitude a of oscillations is called a resonance curve. It has the following key features:
- 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
The maximum amplitude of the oscillations occurs when the driving frequency is equal to the natural frequency of the oscillator
Damping reduces the??
- 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
As the degree of damping is increased, the resonance graph is altered in the following ways:
- 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
As damping is increased, resonance peak lowers, the curve broadens and moves slightly to the left
