Simple Harmonic motion Flashcards
What is simpler harmonic motion?
Simple harmonic motion (SHM) is a specific type of oscillation
An oscillation is said to be SHM when:
The acceleration is proportional to the displacement
The acceleration is in the opposite direction to the displacement
What some examples of oscillators that undergo SHM?
Examples of oscillators that undergo SHM are:
The pendulum of a clock
A mass on a spring
Guitar strings
The electrons in alternating current flowing through a wire
An object in SHM will contain what force?
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
What is the equation for acceleration in SHM?
a = -(⍵^2)x
a = acceleration (m s-2)
⍵ = angular frequency (rad s-1)
x = displacement (m)
What does the equation for acceleration in SHM demonstrate?
The equation demonstrates:
The acceleration reaches its maximum value when the displacement is at a
maximum ie. x = A (amplitude)
The minus sign shows that when the object is displaced to the right, the direction of
the acceleration is to the left and vice versa (a and x are always in opposite
directions to each other)
What does the graph for acceleration against displacement look like and what are some key features of the graph?
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 −A and +A
Graph provided in document
What is the solution to the SHM acceleration equation?
x=Acos(⍵t)
x=Asin(⍵t)
A = amplitude (m)
t = time (s)
⍵ = angular frequency (rad s-1)
x = displacement (m)
When should you use cos or sin when solving for acceleration?
x = A cos (⍵t)
Where:
A = amplitude (m)
t = time (s)
This occurs when:
An object is oscillating from its amplitude position (x = A or x = −A at t = 0)
The displacement will be at its maximum when cos(⍵t) equals 1 or −1, when x = A
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
If an object is oscillating from its equilibrium position (x = 0 at t = 0) then the displacement
equation will be:
x = A sin (⍵t)
The displacement will be at its maximum when sin(⍵t) equals 1 or −1, when x = A
This is because the sine graph starts at 0, whereas the cosine graph starts at a maximum
Draw the graphs for sin and cos
Graphs provided in the document
What happens to the speed of an object in SHM and when is it at its greatest?
The speed of an object in simple harmonic motion varies as it oscillates back and forth
Its speed is the magnitude of its velocity
The greatest speed of an oscillator is at the equilibrium position ie. when its displacement x
= 0.
What is the equation for velocity in SHM?
v=±⍵((A^2-x^2))^1/2
v = velocity (m s-1)
A = amplitude (m)
± = ‘plus or minus’. The value can be negative or positive
⍵ = angular frequency (rad s-1)
x = displacement (m)
What are all undamped SHM graphs represented by?
All undamped SHM graphs are represented by periodic functions
This means they can all be described by sine and cosine curves.
Draw the graph for an object in SHM where when x=0 and T=0
Graphs provided in the document
What are some key features of the displacement graph?
Key features of the displacement-time graph:
The amplitude of oscillations A 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
What are some key features of the velocity graph?
Key features of the velocity-time graph:
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:
An oscillator moves the fastest at its equilibrium position
Therefore, the velocity is at its maximum when the displacement is zero
What are some key features of the acceleration graph?
Key features of the acceleration-time graph:
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°
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 maximum value of the acceleration is when the oscillator is at its maximum
displacement.
When will the minimum and maximum speed during SHM occur?
The speed v of an oscillator will vary in SHM. It is:
Maximum at the equilibrium position (x = 0)
Zero at the amplitude (x = A)
What is the equation for maximum velocity for SHM?
vmax = ωA
vmax = maximum velocity (m s-1)
ω = angular frequency (rad s-1)
A = amplitude (m)
What is the graph for v against t?
Graph provided in the document
When does the maximum and minimum acceleration occur?
The acceleration a of an oscillator will also vary in SHM. It is:
Maximum at the amplitude (x = A)
Zero at the equilibrium position (x = 0)
What is the equation for maximum accleration?
amax = (ω^2)A
amax = maximum acceleration (m-s2)
ω = angular frequency (rad s-1)
A = amplitude (m)
What is the equation for a period mass spring system?
T=2π(m/k)^1/2
T = time period (s)
m = mass (kg)
k = spring constant (N m-1)
What is the equation for the time period of a simple pendulum?
T=2π(L/g)^1/2
T = time period (s)
L = length of string (m)
g = gravitational field strength (N kg-1)
Draw a diagram for the simple pendulum
Diagram provided with document
What is the limitation of using the equation for the time period for a simple pendulum
The limitation with this formula is that it only works for small angles from the equilibrium
point (~10°)
What are some examples where you might be expected to treat them as a harmonic oscillator?
Some other examples include:
A bungee jumper
An acrobat on a trapeze
A swing
A ball in a concave dish
Oscillating platforms
A liquid in a U-tube
What are the two forms energy is exchanged between in SHM and examples for one of them
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)
When is kinetic energy at a maximum?
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)
When is potential energy at a maximum?
Speed v is 0 (and kinetic energy is 0) at maximum displacement x = A, so:
The potential energy is at a maximum when the displacement (both positive
and negative) is at a maximum x = A (amplitude)
What can you say the total energy within a simple harmonic system is?
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.
Draw the graph between Kinetic energy and potential energy against time and key features of the graph
The key features of the energy-time graph are:
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
Graph provided in the document
How many cycles does kinetic and potential energy go through one time period of oscillation and why?
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)
Draw the graph of Ek, Ep and Et against displacement and key features of the graph
The key features of the energy-displacement graph are:
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 x = A, 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 x = A, and
maximum at the equilibrium position x = 0
This is represented by an ‘n’ shaped curve
The total energy is represented by a horizontal straight line above the curves.
Graphs provided in the document
What are the resistive force acting on SHM systems called?
In practice, all oscillators eventually stop oscillating
Their amplitudes decrease rapidly, or gradually
This happens due to resistive forces, such as 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.
Definition for damping
The reduction in energy and amplitude of oscillations due to resistive forces
on the oscillating system.
What must you remember about frequency and damping?
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
For example, a child on a swing can oscillate back and forth once every second, but
this time remains the same regardless of the amplitude.
What are the different 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
What is 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
For example, a swinging pendulum decreasing in amplitude until it comes to a stop.
What is the graph for light damping and key features?
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
Graph provided in the document
What is 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.
What is the graph for critical damping and key features?
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
Graph provided in the document
What is 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
What is the graph for heavy damping and key features?
Key features of a displacement-time graph for a heavily damped system:
There are no oscillations. This means the displacement does not pass zero
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
Graph provided in the document
What are free oscillations?
Free oscillations occur when there is no transfer of energy to or from the surroundings
This happens when an oscillating system is displaced and then left to oscillate
In practice, this only happens in a vacuum. However, anything vibrating in air is still
considered a free vibration as long as there are no external forces acting upon it
Therefore, a free oscillation is defined as:
An oscillation where there are only internal forces (and no external forces)
acting and there is no energy input
A free vibration always oscillates at its resonant frequency.
What are forced oscillations?
In order to sustain oscillations in a simple harmonic system, a periodic force must be applied
to replace the energy lost in damping
This periodic force does work on the resistive force decreasing the oscillations
It is sometimes known as an external driving force
These are known as forced oscillations (or vibrations), and are defined as:
Oscillations acted on by a periodic external force where energy is given in
order to sustain oscillations
Forced oscillations are made to oscillate at the same frequency as the oscillator creating the
external, periodic driving force
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.
What is natural frequency and driving frequency and how does this link to resonance?
The frequency of forced oscillations is referred to as the driving frequency (f) or the
frequency of the applied force
All oscillating systems have a natural frequency (f0), this is defined as this is the frequency
of an oscillation when the oscillating system is allowed to oscillate freely
Oscillating systems can exhibit a property known as resonance
When the driving frequency approaches the natural frequency of an oscillator, the system
gains more energy from the driving force
Eventually, when they are equal, the oscillator vibrates with its maximum amplitude,
this is resonance.
What is the definition of resonance?
Resonance is defined as:
When the frequency of the applied force to an oscillating system is equal to
its natural frequency, the amplitude of the resulting oscillations increases
significantly.
Example of resonance?
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. This frequency at which this push happens is the driving frequency
When the driving frequency is exactly equal to the natural frequency of the swing
oscillations, resonance occurs
If the driving frequency does not quite match the natural frequency, the amplitude
will increase but not to the same extent as when resonance is achieved
How is energy transferred during resonance?
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.
What is the graph of driving frequency against Amplitude called and key features as well as draw
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
Graph provided in document
What are the effects of damping on resonance?
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 of the oscillator 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
Therefore, damping reduced the sharpness of resonance and reduces the amplitude at
resonant frequency
What does the resonance curve look like with different damping?
Graph provided in document
When does resonance happen and some examples?
Resonance occurs for any forced oscillation where the frequency of the driving force is equal
to the natural frequency of the oscillator
Examples include:
An organ pipe, where air resonates down an air column setting up a stationary wave
in the pipe
Glass smashing from a high pitched sound wave at the right frequency
A radio tuned so that the electric circuit resonates at the same frequency as the
specific broadcast
What is a mechanical system meant to show resonance. what happens and draw a diagram
A mechanical system commonly used to show resonance is Barton’s pendulums
A set of light pendulums labelled A-E are suspended from a string
A heavy pendulum X, with a length L, is attached to the string at one end and will act
as the driving pendulum
When pendulum X is released, it pushes the string and begins to drive the other pendulums
Most of the pendulums swing with a low amplitude but pendulum C with the same length
L has the largest amplitude
This is because its natural frequency is equal to the frequency of pendulum X (the
driving frequency)
The phase of the oscillations relative to the driver are:
Pendulums E and D with lengths < L are in phase
Pendulum C with length = L is 0.5π out of phase
Pendulums B and A with lengths > L are π out of phase
Graph provided in the document
