G484 - Circular Motion and Oscillations Flashcards
Degrees to Radians
x (2π/360)
Radians to Degrees
x(360/2π)
State, in terms of force, the conditions necessary for an object to move in a circular path at constant speed
Resultant (centripetal force)
Perpendicular to the direction of motion
Centripetal Acceleration
Directed towards the centre of the circle
Direction of motion is always changing so velocity is always changing even if speed remains constant
Acceleration is the rate of change of velocity
Describe how a mass creates a gravitational field in the space around it
All objects with mass have a gravitational field
The field spreads out into the space around the object in all directions
As a result any other object with mass that is in the field will experience a force of attraction
Gravitational Field Strength
Force per unit mass
Close to the earth’s surface, gravitational field strength is approximately equal to the acceleration of free fall
g = -GM/r²
Newton’s Law of Gravitation
The force between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between them
F = - GMm/r²
Properties of Gravitational Fields
Unlimited range
Always attractive
Effect all objects with mass
Becomes zero at centre of mass of a sphere as the mass surrounding that point is exerting a force in every direction so the resultant is 0
Period
of an object describing a circle
Time taken for the object to complete one circular path
Kepler’s Third Law
T² ∝ r³
T² = (4π²/GM)r³
Derive T² = (4π²/GM)r³ from first principles
v² = GM/r , v = 2πr/T
(2πr/T)² = GM/r 4π²r²/T² = GM/r 4π²r³ = GMT² T² = (4π²/GM)r³
Geostationary Orbit
24 hour time period
Equatorial orbit - orbits over the equator
Orbits in the same direction as the earth’s rotation
Satellite appears to remain stationary in the sky above a particular point
Radian
The angle subtended by an arc of the circumference equal to the radius
Displacement
Distance form the equilibrium position
Amplitude
Maximum displacement from the equilibrium position
Period
Time taken for one complete oscillation
Frequency
Number of oscillations completed in one second
Angular Frequency
2π x frequency
OR
2π / period
Phase Difference
The difference between the pattern of vibration of two points/two waves where one leads or lags begins the other
Simple Harmonic Motion
Acceleration is directly proportional to displacement but in the opposite direction i.e. towards the equilibrium position
An object is in S.H.M. if a = -(2πf)² x
Simple Harmonic Motion Velocity Equation
v = 2πf √(A²-x²)
Simple Harmonic Motion - Period and Amplitude
The period of an object with simple harmonic motion is independent of its amplitude
S.H.M. Displacement Graph
Sine curve
S.H.M. Velocity Graph
Cosine graph
S.H.M. Acceleration Graph
Negative Sine Curve
Time Period - Pendulum Equation
T = 2π√(l/g)
Time Period - Spring Equation
T = 2π √(m/k)
k = spring constant
In Phase
Oscillations with the same frequency that reach amplitude at the same time
Out of Phase
Oscillations with the same frequency that reach amplitude at different times
Damping
Decrease in amplitude of oscillations over time as a result of a resistive force
Light Damping
Gradual decrease in amplitude and increase in time period
Heavy Damping
Amplitude of oscillation decreases to 0 rapidly and overshoots the equilibrium position before coming to rest
Critical Damping
Amplitude decreases to 0 in the shortest possible time
Over damping
A very slow return to the equilibrium position
Light Damping Example
Sound level meters
Show rapid fluctuations in sound intensity
Heavy Damping Example
Car Fuel Gauges
So that the pointer does not oscillate
Ignores small transient changes in the fuel level in the tank
Critical Damping Example
-Pointer instruments e.g. voltmeter, ammeter
Pointer reaches correct position after a single oscillation
Prevents oscillation around the actual value
-car suspension system
Passengers don’t bounce up and down when the car passes over bumps
Natural Frequency
The frequency that an object will vibrate at freely after an initial disturbance
Forced Oscillations
When a periodic force is applied to an oscillating object, the object is forced to oscillate at the same frequency as the driver of the periodic force
Resonance
Occurs when as oscillating system is forced to vibrate at a frequency close to its natural frequency
Amplitude of vibration increases rapidly and becomes maximum when the frequency of the force us equal to the natural frequency
Useful Applications of Resonance
Driver -> Resonator Cooking - microwaves -> water molecules MRI - radio waves -> nuclei/protons Woodwind Instruments - reed -> air column Brass Instruments - lips -> air column
Problems Due to Resonance
Driver -> Resonator
Engine Vibrations -> Car Windows
Wind -> Bridges
Earthquake - Ground Vibrating -> Buildings
Resonance Without Damping
- amplitude and energy of a resonating system would increase continually
- in practise this can never happen as there is always some degree of damping
Resonance With Damping
-the amplitude and energy would increase until energy was being dissipated at the same rate as it was being supplied
