Simple Harmonic Motion Flashcards
Define the amplitude of an oscillating object
The amplitude is the maximum displacement of the oscillating object from the equilibrium
Give the conditions required for an oscillating object to be described as freely oscillating
- Constant amplitude
- no frictional forces are present
Define the time period T, of an oscillating object
The time period is the time for one complete cycle of oscilations
Define the frequency of oscillations, f
The frequency of oscillations is the number of cycles per second made by an oscillating object
Give the unit for frequency
Hz
Give the equation linking time period T, to frequency f
T = 1 / f
Give the angular frequency (velocity) , for and object with time period T and give its units
ω = 2π / T = 2πf
Units : rads⁻¹
Give the phase difference for two objects oscillating at the same frequency
Phase difference (in radians) = 2πΔt / T
where Δt is the time between successive instants when the two objects are at maximum displacement in the same direction
For an object oscillating at a constant frequency, describe the differences of the graphs of:
i) displacement vs. time
ii) velocity vs. time
iii) acceleration vs. time
i) If the object is started with maximum positive displacement, after T/4 it will be at minimum displacement, and at T/2 it will be at maximum negative displacement
ii) the object will start with 0 velocity, but will be at maximum negative velocity at T/4 and at maximum positive velocity at 3T/4
iii) the object will start with maximum negative acceleration, and will have 0 acceleration at times T/4 and 3T/4
Define simple harmonic motion
oscillating motion in which the acceleration is proportional to, but always in the opposite direction to the displacement
Give the acceleration for an object moving with simple harmonic motion
a = - (2πf)²x = - ω²x
where x is the displacement
Give the relationship between the time period, T, and the amplitude of the oscillations
The time period is independent of the amplitude of the oscillations
Give the relationship between the maximum displacement and the amplitude of an oscillating object
x(max) = ±A
Describe an experiment to compare simple harmonic motion with circular motion
Place a simple pendulum above a turntable with a bob on it in front of a projector. Given that the initial maximum displacement is equal to the radius of the turntable, and that the turntable has the same time period as the pendulum, the shadows of the balls on the screen will stay together as they move
For a ball on top of a turntable rotating with a constant frequency in front of projector, give the equation for the component of the acceleration of the ball which is parallel to the screen (aₓ)
aₓ = acosθ = - (2πf)²rcosθ = - (2πf)²x
For an object at displacement x = +A at t = 0, if it starts with zero velocity, give the equation for its displacement at time t
x = Acos(2πft)
What determines the frequency of oscillation of a loaded spring?
The mass of the load and the strength of the spring
Give the effect on the time period of adding extra mass to an oscillating spring
It would increase the time period (because the inertia of the system would increase so at a given displacement, the load would be slower than if the mass had not been added)
Give the effect on the time period of using weaker springs for the same mass load
It would increase the time period (because the restoring force on the load at any given displacement would be less so the trolley’s acceleration and speed would be less)
Give the equation for the restoring force of a loaded spring
ΔT = - kx
where k is the spring constant, x is the displacement and T is the change in tension of the spring
Give the equation for the acceleration of a loaded spring in terms of its resolving force
a = restoring force / mass = - kx / m
Give the equation for the acceleration of a loaded spring showing that it moves in simple harmonic motion
a = - (2πf)²x
Rearrange (2πf)² = k / m to give the time period of oscillation when the spring constant and mass of the loaded spring is known
(2πf)² = k / m f = (1 / 2π )√(k / m) T = 1 / f T = 2π √(k / m)
When timing oscillations, what would you use to show the centre of oscillations?
A fiducial marker
For a simple pendulum, give the equation for the time period, T
T = 2π √(L / g)
where L is the length of the pendulum
As the bob of a simple pendulum passes through the equilibrium, the tension directly upwards is Tₓ. Give the resultant force of the bob at this instant
Tₓ - mg = mv² / L
Give the acceleration for a simple pendulum when the bob is at angle θ
a = F / m = -mgsinθ / m = -gsinθ
For a freely oscillating object, give the equation for the potential energy at displacement x, and describe the shape of the graph when it oscillates from amplitude -A to +A
Eᴘ = ½kx²
It is parabolic
For a freely oscillating object, give the equation for the kinetic energy at displacement x, and describe the shape of the graph when it oscillates from amplitude -A to +A
Eᴋ = Eᴛ - Eᴘ = ½k(A² - x²)
It is an inverted parabola
Define dissipative forces
The force causing the amplitude to decrease by dissipating the energy of the system to the surroundings as thermal energy
How is the motion of oscillations said to be if a dissipative force is present?
Damped
Define light damping
The amplitude of oscillation gradually decreases, reducing by the same fraction each cycle
Define critical damping
Where the damping is enough to stop the system oscillating after it has been displaced and released from equilibrium.
The oscillating object returns to equilibrium with the shortest possible time without overshooting
Define heavy damping
Where the damping is so strong that the displaced object returns to equilibrium much more slowly than if the system is critically damped. No oscillating motion occurs
Explain how the components of a car’s suspension system works
A suspension spring is present to reduce the forces of impact from jolts.
An oil damper is fitted with each spring to prevent the chassis from oscillating long after the force of impact
Define a periodic force
A force which is applied at regular intervals
Define natural frequency
The frequency at which a system oscillates when not subjected to a periodic force
Define forced oscillations
When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force.
The system undergoes forced oscillations
Define resonance
When the applied frequency is equal to the natural frequency of the oscillating system the amplitude of oscillations becomes very large.
At resonance, what is the phase difference between the displacement and the periodic force?
½π
At resonance, what is the phase relationship between the periodic force and the velocity of the oscillating system?
It is exactly in phase
What effect does damping have on the amplitude of an oscillating system in resonance?
The lighter the damping, the larger the amplitude becomes
For a system in resonance, as the applied frequency becomes increasingly larger than the natural frequency of the oscillating system, what happens to:
i) the amplitude of oscillations?
ii) the phase difference between the displacement and the periodic force?
i) the amplitude of oscillations decreases more and more
ii) the phase difference increases from ½π until the displacement is π radians out of phase with the periodic force
Explain Barton’s pendulum
- Multiple simple pendulums of different length are set up on a piece of wire. A driver pendulum has the same length as one of the other pendulums.
- The driver pendulum is displaced and allowed to oscillate perpendicular to the plane of the pendulums at rest.
- The effect of the oscillation is transmitted along the support thread, subjecting the others to forced oscillations.
- The pendulum with the length the same as the driver pendulum will begin to resonate since it has the same natural frequency.
- The response of each other pendulum depends on how close its length is to the driver pendulum
