SHM/Circular motion

Significance of banked tracks

Horizontal component of the normal contact force, which provides extra centripetal force

Enables objects to complete circle at greater speed without skidding

Banked tracks equation

v² = rgtantheta

No friction

Where r is radius of circle, g acceleration due to gravity, theta slope of angle

Define shm

The relationship between the acceleration of an object and it's displacement

Conditions for shm

Acceleration is directly proportional to displacement

Acceleration is in the opposite direction to displacement

Features of time period in shm

Independent of amplitude

Increases as mass increases but not directly proportional

Amplitude definition

Maximum displacement from equilibrium position

When does maximum speed occur

When an object passes through the equilibrium position

When does maximum acceleration occur

At the extremes of the oscillation

For pendulum shim restoring force

F = mgsintheta. = mgtheta small angle approximation

Free oscillation definition

Energy is conserved (no friction) so amplitude is constant

Damped oscillation definition

Energy is lost to their surroundings due to friction, air resistance, etc.

Damping definition

Resistive force which acts in the opposite direction to velocity to reduce amplitude

Natural frequency definition

Frequency the system oscillates at with no energy input

Forced vibration

Oscillation caused by a periodic driving force

Free vibration

When a system is displaced and left to oscillate

Resonance

Occurs when driving oscillation is at the same frequency as the natural frequency

Leads to sharp increase in amplitude = large amplitude

When forced frequency ≠ natural frequency

Forced vibration at given frequency and small amplitude

The effect of damping on resonance

No/little damping, largest value for max amplitude occurs when frequency driving oscillation = natural frequency

Resonant frequency decreases as the degree of damping increases

The resonance curve becomes less sharp as the damping is increased

Phase difference for driving frequency much less than natural frequency

Phase difference = 0

Amplitude ≈ of applied

Phase difference for driving frequency = natural f

Phase difference = 90

Phase difference for driving f much larger than natural f

Phase difference = 180

Reducing effects of resonance

Extra damping to absorb energy and reduce amplitude

Change natural frequency (e.g. adding mass)

Light damping

Takes a long time for the amplitude to decrease to 0. System oscillates at natural f

Critical damping

Shortest time for amplitude to decrease to 0

Heavy damping

Takes long time for amplitude to decrease to 0. No oscillating motion occurs

Conditions in time period equation of a pendulum

Oscillations with a small amplitude

Angular displacement <10°