Simple harmonic and circular motion Flashcards
What are the criteria for SHM?
An object is displaced by some distance from a starting position. A restoring force acts to return the object to its starting position. The acceleration is always opposite to the direction of displacement and proportional to displacement.
Graph of acceleration against displacement.
Straight line with negative gradient through origin.
Formula for angular frequency
2πf
Or 2π/T
Graph of displacement against time
Cos graph
Graph of velocity against time
-sin graph
Graph of acceleration against time
-cos graph
When t=0 what is the displacement equal to?
The amplitude
What factors determine frequency of oscillations in a mass spring system?
Extra mass increases inertia of system so moves slower and so frequency lower. Weaker springs reduce restoring force so acceleration and speed both less so frequency less.
Describe energy changes during one oscillation of a mass spring system
Elastic potential energy to kinetic energy to gravitational potential energy to kinetic energy to elastic potential energy.
Graph of KE against time
Sin wave but X axis connects troughs
Graph of GPE against time
Cos graph but X axis connects troughs
Graph of total energy against displacement
KE is n shape. GPE is u shape. Both touch but don’t cross X axis.
Graph of amplitude against driving frequency
Volcano shape with peak at resonant frequency.
How does damping affect amplitude against driving frequency graph?
Peaks are lower and move slightly left.
When does the formula for time period apply to a pendulum?
When the angle is less than about 10 degrees
Forced vibrations
Oscillations of a system subjected to an external periodic force.
Free vibrations
Oscillations where there is no damping and no periodic force acting on the system so the amplitude is constant.
Formula for phase difference
2pi x t/T
Damping
Where the amplitude of an oscillating system decreases because of dissipative forces acting on the system which dissipate energy from the system to the surroundings.
Light damping
Amplitude gradually decreases, reducing by the same fraction each cycle. The time period is independent of the amplitude.
Critical damping
Just enough energy to stop system oscillating after it has been displaced from equilibrium and released. The system returns to equilibrium in the shortest possible time without overshooting. Example is car suspension.
Heavy damping
Damping is so strong, the displaced object returns to equilibrium much more slowly than for critical damping. No oscillating motion occurs. Example is mass on spring in thick oil.
Periodic force
A force that varies regularly in magnitude with a definite time period.
Resonance
The amplitude of vibration of an oscillating system subjected to a periodic force is largest when the periodic force has the same frequency as the resonant frequency of the system. System vibrates such that its velocity is in phase with the periodic force.
Resonant frequency
The frequency of an oscillating system in resonance.
How does damping affect resonance?
Lighter damping means maximum amplitude larger and resonant frequency is closer to natural frequency of the system.
What happens as the applied frequency becomes increasingly larger than the resonant frequency of a mass spring system?
Amplitude of oscillations decreases more and more, phase difference between displacement and periodic force increases from 1/2 pi to pi.
What is the resonant frequency of a system with little or no damping?
It’s natural frequency.
What is uniform circular motion?
Motion of an object moving at a constant speed along a circular path.
Angular displacement
The angle an object in circular motion turns through. Equals 2pift or 2pit/T
Angular speed
The rate of change of angular displacement of an object in circular motion.
Why is an object in circular motion always accelerating?
Its velocity is continually changing direction so the velocity is changing.
What is centripetal acceleration and why does it exist.
It is the acceleration of an object in circular motion towards the centre of the circle. The change in direction of its velocity is always towards the centre of the circle so is accelerating in that direction.
Centripetal force definition and examples
The resultant force on an object that moves along a circular path. Acts towards centre of circle. For satellites in orbit, centripetal force is gravity. For object attached to string, centripetal force depends on tension and weight.
Forces in circular motion: over a hill
The weight minus the contact force is the centripetal force. If the car’s velocity is too high, it will lose contact with the road and lose contact force.
Forces in circular motion: roundabout
Centripetal force is provided by sideways force of friction. There is a limiting force of friction depending on the coefficient of friction so if velocity gets too high the car will skid.
Forces in circular motion: banked track (eg nascar)
Like roundabouts only limiting velocity before skidding is higher as there is a component of the contact force acting horizontally. For no sideways friction, it is this horizontal component that is the centripetal force.
Object in circular motion attached to string
Tension in string is force that changes. At object’s lowest point, tension is centripetal force plus weight. At highest point, tension is centripetal force minus weight.
Why is a fiduciary marker placed at equilibrium position?
This is where the mass is moving with greatest speed so is near that point for the least time of any points so it reduces the error when judging when an oscillation is complete.
How to find g using a pendulum
Vary the length of pendulum and time a number of oscillations. Use this the find the time period of one oscillation for different lengths. Use gradient of time period against root length to find g.
How to find spring constant using mass-spring system
Vary mass on spring. Time a number of oscillations. Find time period for one oscillation for each mass. Use gradient of graph of time period against root mass to find spring constant.
