6.1 periodic motion

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Centripetal acceleration
Due to newtons second law, an object travelling in a circle has a changing velocity even though its speed is constant as it is changing direction
Acceleration is the rate of change of velocity so the car is accelerating despite the constant speed
The centripetal acceleration always acts towards the centre of the circle
Acceleration and force act in the same direction
F = ma
$a=$ $v^2/r =$ $w^2r$
Angular speed
The angle turned per unit of time
w=x/t
rad s-1
Linear speed = tangential velocity
w=vr
Frequency = number of complete revolutions per second
Period = time taken for a complete revolution
f=1/T
For one revolution an object turns through 2π radians
w=2π/T = 2πf
Radians
Angle in radians = arc-length divided by circle radius
For a complete circle, the arc length is the circumference
Divided by radius gives 2π therefore 2π radians in a complete circle
Centripetal force
From newtons laws, if there's a centripetal acceleration there must be a centripetal force acting towards the centre of the circle
This force keeps the object travelling in a circle
$F=$ $mv^2/r =$ $mw^2r$
e.g. questions
If an object is on top of a curve:
Centripetal force and weight acting down towards the centre
Reaction/normal force acts against a surface, up, away from the centre
$mv^2/r=$ $mg-N$

If an object is on the bottom of a curve:
Centripetal force and reaction/normal force acting up towards the centre
Weight acts down away from centre
$mv^2/r=$ $N-mg$

If an object is underneath the top of a curve:
Centripetal force, reaction/normal force and weight acting down towards the centre
$mv^2/r=$ $mg+$ $N$
Conical pendulum:
(Resultant tension)2 = (horizontal tension)2 + (vertical tension)2
Centripetal force acts towards the centre of the circle formed by swinging the object
Weight acts down towards the centre of the earth
$mv^2/r=$ $sin(x)*$ $T_h$
$mg=$ $cos(x)*$ $T_v$
$mv^2/rmg=$ $Tsin(x)/Tcos(x)$
$v^2/rg=$ $tan(x)$
Banked curve:
Weight acts down towards the centre of the earth
Friction acts parallel to the slope
Centripetal force acts towards the centre
Normal/reaction force acts perpendicularly away from the slope
$mv^2/r=$ $N*$ $sin(x)$
$mg=$ $N*$ $cos(x)$
$mv^2/rmg=$ $Tsin(x)/Tcos(x)$
$v^2/rg=$ $tan(x)$
Characteristics of simple harmonic motion
An object moving with simple harmonic motion oscillates to maximum and minimum displacement passing through a midpoint of no displacement
There is always a restoring force pulling or pushing the object towards this point of zero displacement
The size of the restoring force is directly proportional to the displacement
acceleration is directly proportional to displacement
As the restoring force causes acceleration towards the midpoint
Condition for SHM
An oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint and is directed towards the midpoint
a ∝ -x a=-kx
SHM equations
$a=$ $-\omega^{2}x$
Acceleration is always in the opposite direction of displacement
$a_{max}=$ $\omega^{2}A$
$v=$ $+$ $/-\omega\sqrt{A^{2}-x^{2}}$
Velocity is positive when moving in one direction, and negative in the other
$v_{max}=$ $\omega A$
$x=$ $A\cos\left(\omega t\right)$
Displacement varies with time so timing needs to start when the pendulum is at maximum displacement
Graphs
Displacement-time graph is a cosine graph
Maximum value A - the amplitude
Velocity is a negative sine graph
It is the gradient of a displacement-time graph - the derivative
maximum value $\omega A$
Is a quarter of a cycle in front of the displacement
Acceleration is a negative cosine graph
It is the gradient of a velocity-time graph
Maximum value $\omega^{2}A$
In antiphase with the displacement
Mass-spring system
A force is exerted on a mass when it is pushed away from the point of equilibrium
$F=$ $-kx$
The period of a mass oscillating on a spring is derived from Newton’s second law
$T=$ $2\pi\sqrt{\frac{m}{k}}$
Affected by mass and stiffness of the spring
Simple pendulum
$T=$ $2\pi\sqrt{\frac{l}{g}}$
Affected by the length of the pendulum and gravitational field strength
Energy and restoring force
The type of potential energy depends on what is providing the restoring force
Pendulums - gravitational potential energy
Spring-mass system - elastic potential energy
Restoring force does work on the object as it moves towards the point of equilibrium
Transfers GPE to KE
Moving away from equilibrium
Transfers KE to GPE
At equilibrium, GPE is zero, and KE is maximum
At maximum displacement GPE is maximum, and KE is zero
Mechanical energy - the sum of GPE and KE
Constant - when not damped
Energy transfer of one complete oscillation: GPE to KE to GPE to KE to GPE
Method: mass-spring system
Set up equipment and pull masses down a set amount for the initial amplitude
Let go of masses to allow them to oscillate with simple harmonic motion
Position sensor measures the displacement of mass over time
Connect the position sensor to the computer and create a displacement-time graph for the period
Method: simple pendulum
Attach pendulum to angle sensor connected to computer
Displace the pendulum from the position at an angle less than 10°
Let go of the pendulum to allow oscillation with simple harmonic motion
The angle sensor measures how the bob’s displacement varies with time
Use a computer to plot a displacement-time graph for the period
Calculate the average period over several oscillations to reduce the percentage uncertainty
Change the mass of the bob, amplitude and length of the pendulum to see the effect on the period
Free vibrations
Free vibration - no transfer of energy to or from the surroundings
An object in simple harmonic motion oscillates at its resonant frequency
If no energy is transferred it will keep oscillating at the same amplitude forever
Oscillations in the air are considered free vibrations despite this never happening in practice
Forced vibrations
Forced vibration - when there's an external driving force
The system can be forced to oscillate by a periodic external force
Driving frequency - frequency of external force
If the driving frequency is less than the resonant frequency the two are in phase
The oscillator follows the motion of the driving force
If the driving frequency is greater than the resonant frequency the two are out of phase
At resonance, the phase difference between the driving force and oscillator is 90°
Resonance
Resonance - when driving frequency is similar to resonant frequency the system gains energy from the driving force and vibrates with rapidly increasing amplitude
Damping
Damping forces - frictional forces causing oscillating systems to lose energy to surroundings
Systems are damped to stop oscillating or minimise resonance
Damping - reduces the amplitude of oscillation over time
The heavier the damping the quicker the amplitude is reduced to zero
Damping
Critical damping - reduces the amplitude in the shortest time possible
Car suspension systems and moving coil meters are critically damped so they don't oscillate but return to equilibrium as quickly as possible
Underdamping - doesn't completely reduce the amplitude to zero
Overdamping - more damping than critical damping slowing oscillation and taking longer to return to equilibrium
Plastic deformation of ductile materials reduces amplitude in the same way
As material changes shape it absorbs energy reducing oscillations
Effect of damping on resonance
Lightly damped systems have very sharp resonance peak
Amplitude only increases dramatically when driving frequency is close to resonant frequency
Heavily damped systems have a flatter response
Amplitude doesn't increase near resonant frequency and isn't as sensitive to driving frequency
Examples of damping
Structures are damped to avoid damage from resonance
Taipei 101 - skyscraper using giant pendulum to damp oscillations caused by wind
Damping can improve performance
Loudspeakers - create sound waves in the air which reflect off walls and at certain frequencies produce stationary sound waves causing resonance
Resonance can reduce the quality of the sound so soundproofing is used to absorb the energy and convert it into heat energy