Simple Harmonic Motion (Need to add RP) Flashcards
What is simple harmonic motion
A specific type of oscillation where there is repetitive movement back and forth through an equilibrium position.
What is the time period for each complete vibration in SHM
Time interval is the same
In which direction does the restoring force act
Always directed horizontally or vertically towards the equilibrium position
Relationship between distance from equilibrium and the restoring force
Directly proportional
Examples of SHM
Pendulum of a clock
Child on a swing
Mass on a string
Condition for SHM
The acceleration is proportional to the horizontal or vertical displacement
The acceleration is in the opposite direction to the displacement
a is directly proportional to -x
Acceleration of an object oscillating in SHM =
-(angular velocity^2 x displacement)
State the velocity, acceleration and force when the displacement = +max
Velocity = 0
Acceleration = -max
Force = -max
State the velocity, acceleration and force when the displacement = -max
Velocity = 0
Acceleration = +max
Force = +max
State the velocity, acceleration and force when the displacement = 0
Velocity = max
Acceleration = 0
Force = 0
What does the graph of acceleration against displacement look like
Straight line through the origin sloping downwards (like y = -x)
Velocity in terms of displacement
Rate of change of displacement
Acceleration in terms of velocity
Rate of change of velocity
What does a displacement-time graph look like, when oscillation start from equilibrium
Sine curve
What does velocity-time graph look like, when oscillation start from equilibrium
Cosine curve
What does acceleration-time graph look like, when oscillation start from equilibrium
negative sine curve
How are all the graphs derived
v-t graph derived from gradient of x-t graph
a-t graph derived from gradient of v-t graph
Equation for restoring force
Force = -kx
Where k is a constant.
What happens to the time period of the oscillation as spring constant increases
The spring will be stiffer so exerts a larger restoring force and the time period of the oscillation will be shorter
What does the time period of a pendulum depend on
Gravitational field strength, therefore would be different on different planets
sin theta in a pendulum =
approximately theta as the formula is limited to small angles (smaller than 10 degrees)
What is the restoring force of the pendulum
The weight component acting along the arc of the circle towards the equilibrium position and is resolved to act act at and angle theta to the horizontal x.
Why is it assumed the restoring force in SHM in a pendulum acts along the horizontal
Because of small angle approximation
Which equation do you use for situations such as liquid in a U-tube
The same as the equation for a simple pendulum as it can be modelled in the same way
What energies are involved in the swinging of a pnedulum
GPE and KE
What energies are involved in the horizontal oscillation of a mass on a spring
Elastic potential and Kinetic energy
In a mass-spring system where is the max elastic potential energy
When the spring is stretched beyond its equilibrium position
What happens to KE when the mass in a mass-spring system is released
Mass moves back towards equilibrium position and it accelerates and causes KE to increase
When is KE at its max in a mass-spring system
At equilbrium position
When is EPE at its minimum in a spring system
At equilibrium position
What happens to KE and EPE once past the equilibrium position in a mass spring-system
KE decreases and EPE increases
When is the GPE max in a simple pendulum
At the amplitude top of the swing
What happens to KE when the pendulum is released
Pendulum moves back towards the equilibrium position and accelerates so the KE increases
What happens to the GPE as the height of the pendulum decreases
GPE decreases
What happens to both GPE and KE once the mass has passed the equilibrium position
KE decreases and GPE increases
What happens to the total energy of a simple harmonic system
It always remains constant and is equal to the sum of KE and GPE/EPE
Key features of an energy-displacement graph
Potential energy is at max at the amplitude and 0 at the equilibrium and is represented by U shape
KE is 0 at amplitude and max at the equilibrium position and is represented by an n shape
Total energy is represented by a horizontal straight line
Key features of an energy-time graph
KE and potential energy are always in complete opposite positions. e.g. max KE = min PE
What is damping (simple)
A resistive force that causes an oscillating object to stop oscillating such as friction and air resistance
What are the 3 types of damping
Light damping
Heavy damping
Critical damping
What direction does damping force act in
Opposite to velocity
It is proportional to negative velocity
Damping (definition)
The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system
How long does a damping force last
Until the oscillator comes to rest at the equilibrium position
What happens to the frequency of damped oscillations as the amplitude decreases
The frequency DOES NOT CHANGE as the amplitude decreases
Light damping characteristics
The amplitude does not decrease linearly, but exponentially with time
Lightly damped oscillating will oscillate with gradually decreasing amplitude
Time period is the same
Features of a displacement-time graph for a lightly damped system
Many oscillations represented by a sine or cosine curve with gradually decreasing amplitude over time
Height of curve decreasing in both positive and negative displacement values
Amplitude decreases exponentially
Time period is the same and peaks and troughs are equally apart
Critical damping characteristics
A critically damped oscillator returns to rest at its equilibrium position in shortest time possible without oscillating
Features of a displacement-time graph for a critically damped system
System does not oscillate so displacement falls to 0 straight away
Fast decreasing gradient until it reaches the x axis
When oscillator reach equilibrium, the graph is a horizontal line at x=0 for remaining time
Heavy damping characteristics
Takes a long time to return to equilibrium position without oscillating
System returns to equilibrium more slowly than critical damping
Key features of a displacement-time graph for heavily damped system
No oscillations so displacement does not pass zero
Slow decreasing gradient until it reaches x axis
Once oscillator reaches equilibrium position, the graph remains a horizontal line
Difference between resistive force and restoring force
Resistive force opposes the motion/velocity of the oscillator and causes damping
Restoring force is what brings the oscillator back to equilibrium position
When do free oscillations occur
When there is no transfer of energy to or from the surroundings.
This happens when an oscillating system is displaced and left to oscillate
Free oscillation
An oscillation when there are only internal forces and no external forces acting and there is no energy input
Forced oscillations
Oscillations acted on by a periodic external force where energy is given in order to sustain oscillations
Why must a periodic force be needed to sustain oscillations in a SHS
To replace the energy lost in damping. The periodic force does work on the resistive force decreasing the oscillations.
What frequency does a free vibration oscillate at
Its resonant/natural frequency
What frequency do forced oscillations vibrate at
The same frequency as the oscillator creating the external, periodic driving force
Natural frequency (f0)
The frequency of an oscillation when the oscillating system is allowed to oscillate freely
Resonance
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
Why is amplitude greatest at resonance
Energy is transferred from the drive to the oscillating system most efficiently therefore the system transfers the max KE possible
Features of a driving frequency against amplitude graph / resonance curve
When f < f0 , the amplitude increases (f is the driving frequency, f0 is natural frequency)
At the peak where f = f0 , the amplitude is at its max - resonance
when f > f0, the amplitude starts to decreases
Effect of damping on resonance
Damping reduces the amplitude of resonance vibrations
Height and shape of curve changes on the degree of damping
Effect of damping on the natural frequency
It remains the same
What happens to resonance graph as the degree of damping increases
Amplitude of resonance vibrations decrease - peak is lower
Resonance peak broadens
Resonance peak moves slightly to the left of the natural frequency when heavily damped
Overall effect of damping on the resonance and amplitude
Reduces sharpness of resonance and reduces amplitude at resonant frequency
What happens to resonant frequency at heavier damping
Resonant frequency becomes slightly less than f0
Examples of where resonance occurs
An organ pipe
Glass smashing from a high pitched sound
