Simple Harmonic Motion Flashcards
Examples of simple harmonic motion
Pendulum swinging
Mass bouncing on a spring
Particle in a solid vibrating backwards and forwards
Steel ball rolling in a curved dish
Two conditions for simple harmonic motion/definition
Acceleration of the object is always directed towards the equilibrium position
Acceleration is always proportional to the displacement of the object from the equilibrium position
How is acceleration related to displacement
a ∝ -x
How is time period related to amplitude
It isn’t
They are independant
Changing the amplitude will not affect the time period of oscillations
Three types of oscillations
Free
Damped
Forced
What is a free oscillation
An oscillation in which there are no external forces acting on the oscillating system
Equilibrium position
Position the object will always return to after oscillations have ceased
Displacement
Distance between the object and the equilibrium position
When is acceleration maximum
When displacement is maximum
a ∝ -x
When is acceleration minimum (zero)
When displacement is zero
a ∝ -x
When is velocity maximum
When displacement is zero
When is velocity minimum (zero)
When displacement is maximum
When is kinetic energy minimum
Maximum acceleration/maximum displacement
Total energy
Constant
Et=Ek+Ep
When is kinetic energy maximum
Minimum acceleration/displacement (zero)
When is potential energy minimum
Acceleration is zero/displacement is zero
When is potential energy maximum
Acceleration is maximum/displacement is maximum
Graph for total energy against time
Straight horizontal line that is positive
Displacement to velocity to acceleration
Differentiate
Differentiate graph using trig
Acceleration to velocity to displacement
Integrate
Integrate graph using trig
Phase relationship between displacement and velocity/velocity and acceleration
π/2
Phase relationship between displacement and acceleration
π
When is x=Acos(wt) used
Displacement
Amplitude
Angular frequency x time to give angular displacement
If x=a when t=0
Must remember w is the angular frequency
What can wt become
wt=2πft=2πt/T
t/T is how far through the cycle it is in radians
Maximum kinetic energy
Hence, maximum potential energy and total energy
1/2 m w^2 A^2
Since m and w are constant
Ek ∝ A^2
When is x=Asin(wt) used
Displacement
Amplitude
Angular frequency x time to give angular displacement
If x=0 when t=0
Must remember w is the angular frequency
What factors affect the time period of a mass spring system
Mass
Spring constant
What factors do not affect the time period of a mass spring system
Amplitude
Acceleration due to gravity
Shape of mass (air resistance)
What factors affect the time period of a simple pendulum
Length of string
Acceleration due to gravity
What must you bear in mind for simple pendulums simple harmonic motion
Only follow it at small amplitudes
A small swing is one in which the angle theta is small enough that sin theta can be approximated to theta when theta is measured in radians
What is the angle of a simple pendulum is greater than 10 degrees
Wont undergo SHM
How can you use a simple pendulum to find gravity
Set it up
Small angle of swing
Vary length of pendulum and record the time period
Repeat and mean
Plot Time period squared on y against length on x
Gravity is equal to 4π^2/gradient
What is a damped oscillation
Oscillating systems energy decreases over time due to an external force acting on the system
Such as friction between two solid bodies
Viscous forces between solid body and a gas or liquid
Consequence of a dampening force
Energy dissipated to surrounding
Amplitude decreases
Time period remains the same
How is the damping force related to velocity
Damping force ∝ - Velocity^2
When is the dampening force greatest
When is it smallest
Max at equilibrium where velocity is max
Minimum/zero at amplitude where velocity is zero
Lightly damped system
Resisting force is small
Energy transferred to surroundings very slowly
System oscillated and the amplitude reduces gradually
Time period is the same
Critical damping/critically damped system
Energy is transferred to surroundings very rapidly
Oscillator does not actually oscillate at all before coming to rest
A quarter of a cycle carried out
Examples of critical damping
Car suspensions where you don’t want oscillations to occur
Moving coil analogue meters where you don’t want oscillations to occur
Heavily damped system
Dampening force is very large System does not oscillate Slowly returns to the equilibrium position System has almost no kinetic energy Only potential energy
Natural frequency
f0
The frequency an object will oscillate at if there are no external forces acting on it
Driving frequency
The frequency of oscillation when a system is being made to oscillate by a periodic external force
Two parts to a forced oscillating system
Driver; the thing providing the external force and input energy
The driven; the part of the system receiving the input of energy and being made to oscillate
Frequency of driver less than natural frequency
Low amplitude oscillations
Similar amplitude to driving force
In phase with the driving force
Frequency of the driver is equal to natural frequency
Resonance occurs
Oscillations of amplitude increase
Much larger than the driving force
π/2 out of phase with the driving force
Frequency of the driver is greater than the natural frequency
Low amplitude oscillations
π out of phase with driving force
Resonance
Drivers frequency is equal to the natural frequency of the system being driven
Resulting in the system oscillating with a large amplitude
Driving force the same direction as velocity
Work done on the system
Increases its energy
Explain the graph for amplitude of driven system against frequency of driver for the types of damping
No damping asymptote at natural frequency and x axis
Light damping asymptote at x
Heavier damping asymptote at x
Heavier the damping the lower the peak amplitude and the smaller the area
Area decreases as damping increases/lines for each type of damping don’t overlap
All cross y intercept at same point and positive non zero
Phase difference f
0
Phase difference f=f0
90°
Phase difference f>f0
180°
Explain Bartons pendulum
Length the same as the driver pendulum: Same frequency so phase difference is 90° and resonance occurs
Length shorter than the driver: Larger frequency so phase difference is 0
Length longer than the driver: Smaller frequency so phase difference is 180°
