Physics - Simple Harmonic Motion

What does amplitude mean?

The amplitude of the oscillations is the maximum displacement of the oscillating object from equilibrium.

What does free vibrations mean?

If the amplitude is constant and no frictional forces are present, the oscillations are described as free vibrations.

What does time period, T, mean?

The time period, T, of the oscillating motion is the time for one complete cycle of oscillation.

What does frequency mean?

The frequency of oscillations is the number of cycles per second made by an oscillating object.

The units are Hz - Hertz

What is phase difference?

For oscillations that are identical but start at a different point in time say Δt later the phase difference is:

(2πΔt)/T

Where T is the time period of the oscillation.

What is the relationship between the displacement - time graph and velocity - time graph of an oscillating object?

The variation of velocity with time is given by the gradient of the displacement - time graph.

What is the relationship between the velocity - time graph and acceleration - time graph of an oscillating object?

The variation of acceleration with time is given by the gradient of the velocity - time graph.

What is the relationship between the displacement - time graph and acceleration - time graph of an oscillating object?

The acceleration is always in the opposite direction to the displacement.

What is simple harmonic motion?

It is defined as oscillating motion in which the acceleration is:

• Proportional to the displacement

• Always in the opposite direction to the displacement

a ∝ -x

What is the equation for acceleration in simple harmonic motion?

Acceleration a = - ω²x

where x is the displacement

where ω is angular frequency

Explain the equation x = Acos(ωt).

For any object oscillating at frequency f in simple harmonic motion, its acceleration is given by a = -ω²x where ω = 2πf. The variation of displacement with time depends on the initial displacement and the initial velocity.

If x = +A when t = 0 and t = 0 and the object has zero velocity at that instant, then its displacement at a later time t is given by x = Acos(ωt).

What determines the frequency of oscillation of a loaded spring?

1. Adding extra mass: This is because the extra mass increases the inertia of the system. At a given displacement, the trolley would therefore be slower than if the extra mass had not been added. Each cycle of oscillation would therefore take longer.

2. Using weaker springs: The restoring force on the oscillating object on the spring at any given displacement would be less. Each cycle of oscillation would therefore take longer.

How do changes in mass and spring constant affect frequency of a small mass m attached to a spring?

• Assuming the spring obeys Hooke’s law, the tension T_s in the spring is proportional to its extension ΔL from its unstretched length. This relationship can be expressed using the equation T_s = kΔL, where k is the spring constant.

• When the object is oscillating and is at displacement x from its equilibrium position, the change of tension in the spring provides the restoring force on the object. Using the equation T_s = kΔL, the change of tention ΔT_s from equilibrium is therefore given by ΔT_{s, =} -kx, where the minus sign represents the fact that the change of tension always tries to restore the object to its equilibrium position.

• So the restoring force on the object is equal to -kx.

• Therefore, the acceleration a = -kx/m

This equation may be written in the form a = -ω²x, where ω² = k/m

Because ω = 2πf, then ω² = k/m

This means f = 1/2π √(m/k)

What is the theory of the simple pendulum?

Consider a simple pendulum that consists of a bob of mass m attached to a thread of length L. If the bob is displaced from equilibrium then released, it oscillated about the lowest point. At displacement s from the lowest point, when the thread is at angle θ to the vertical, the weight mg has components

• mgcosθ perpendicular to the path of the bob

• mgsinθ along the path towards the equilibrium position.

The restoring force F = -mgsinθ, so the acceleration is -gsinθ

This means that

ω² = g/L

Because ω = 2πf

So:

T = 1/f = 2π √(L/g)

What is the simple harmonic motion speed equation?

Using E_k = ½ mv² gives ½ mv² = ½ k(A²-x²), where v is the speed of the object at displacement x.

Because ω² = k/m, the above equation can be written as v² = ω²(A²-x²).

So, v = ±ω√(A²-x²)

Explain the energy-displacement graphs for a simple oscillator.

1. The potential energy curve is parabolic in shape, given by the E_p = ½ kx²

2. The kinetic energy curve is an inverted parabola, given by E_k = E_T - E_p = ½ k(A²-x²)

The sum of the kinetic energy and the potential energy is always equal to ½ kA², which is the potential energy at maximum displacement. This is the same as the kinetic energy at zero displacement. So the two curves add together to give a horizontal line for the total energy.

What does it mean for motion to be damped?

In any oscillating system where friction or air resistance is present the amplitude decreases. The forces causing the amplitude to decrease are described ad dissipative forces because they dissipate the energy of the system to the surroundings as thermal energy. The motion is said to be damped if dissipative forces are present.

What is light damping?

It occurs when the time period is independent of the amplitude so each cycle takes the same length of time as the oscillations die away.

What is critical damping?

Critical dampening is just enough to stop the system oscillating after it has been displaced from equilibrium and released. The oscillating system returns to equilibrium in the shortest possible time without overshooting if the damping is critical. Such damping is important in mass-spring systems such as a vehicle suspension system where you would have an uncomfortable ride if the damping was too light or too heavy.

What is heavy damping?

It occurs when the damping is so string that the displaced object returns to equilibrium much more slowly than if the system is critically damped. No oscillating motion occurs. For example, a mass on a spring in thick oil would return to equilibrium very slowly after being displaced and released.

What is a periodic force?

A force applied at regular intervals.

What is natural frequency?

When the system oscillates without a periodic force being applied to it, the system’s frequency is called its natural frequency.

What are forced vibrations?

When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force. The system undergoes forced vibrations, or forced oscillations, when a periodic force is applied to it.

What is resonance and resonant frequency?

When the periodic force is exactly in phase with the velocity of the oscillating system, and the system is in resonance. The frequency at the maximum amplitude is called the resonant frequency.

What impact does lighter damping have on amplitude and resonant frequency?

The lighter the damping,

• The larger the maximum amplitude because at resonance

• the closer the resonant frequency is to the natural frequency of the system.

As the applied frequency becomes increasingly larger than the resonant frequency of the mass-spring system what happens?

• The amplitude of oscillations decrease more and more,

• The phase difference between the displacement and the periodic force increases from ½ π until displacement is π radians out of phase with the periodic force.

For an oscillating system with little or no damping, at resonance, the applied frequency of the periodic force = the natural frequency of the system.

How are barton’s pendulums an example of resonance?

The driver pendulum D is displaced and released so that it oscillates in a plane perpendicular to the plane of the pendulums at rest. The effect of the oscillating motion of D is transmitted along the support thread, subjecting each of the other pendulums to forced oscillations. Pendulum R responds much more than any other pendulum. This is because it has the same length and therefore the same time period as D. So its natural frequency is the same as the natural frequency of D. Therefore, R oscillates in resonance with D because it is subjected to forced oscillations of the same frequency as its own natural frequency of oscillations. The response of each of the other pendulums depends on how close its length is to the length of D and whether it is shorter or longer than D

How are bridge oscillations an example of resonance?

A bridge span can oscillate because of its springiness and its mass. If a bridge span is not fitted with dampers, it can be made to oscillate at resonance if the bridge span is subjected to a suitable periodic force.

1. A crosswind can cause a periodic force on the bridge span. If the wind speed is such that the periodic force is equal to the natural frequency of the bridge span, resonance can occur in the absence of damping. The dramatic collapse of the Tacoma Narrows Bridge in the US in 1940 was due to such resonance.

2. A steady trail of people in step with each other walking across a footbridge can cause resonant oscillations of the bridge span if there is not enough damping. Soldiers marching in the columns are taught to break out the step with each other when they cross a footbridge, to avoid causing resonance. Soon after it was opened, the Millennium Bridge in London had to be closed and fitted with a more suitable damping system because it swayed in the resonance when people first walked across it.