H2 Physics - Oscillations

State what periodic motion is

It is a motion in which a body continually retraces its path at equal time intervals.

Define equilibrium position

It is the position where the resultant force on the body is zero.

Define restoring force

It is the resultant force that tries to restore the displaced particle to its equilibrium position

Describe what free oscillation is

It is the oscillation which occurs when a body oscillates with no driving and/or resistive forces acting on it. The body oscillates at its natural frequency about the equilibrium position.

Define amplitude

It is the magnitude of the maximum displacement of the particle from its equilibrium position.

Define phase (angle)

The phase of an oscillation that a body is in, is the fraction of an oscillation cycle it is at relative to its starting displacement.

State the formula for angular frequency

ω = 2π/T = 2πf

Define angular frequency

It is the rate of change of phase angle of the oscillation and is equal to the product of 2π and its frequency f

Define phase difference

It is the difference in the positions of two oscillating bodies in their cycles.

State what units phase and phase difference are usually in

Degrees or radian

Define Simple Harmonic Motion (S.H.M)

It is the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point and is always directed towards the point.

State the formula for acceleration of a particle in S.H.M

a = -ω²x

Derive the formula which relates spring constant to angular frequency

• By Hooke's law, F = -kx

• a = F/m = -(k/m)x

• In S.H.M, a = -ω²x

• By comparing, ω² = k/m ⇒ k = ω²m

State when the displacement, velocity and acceleration is at its maximum for a particle in S.H.M

• Displacement: At amplitudes

• Velocity: At equilibrium

• Acceleration: At amplitudes

State when the displacement, velocity and acceleration is at its minimum for a particle in S.H.M

• Displacement: At equilibrium

• Velocity: At amplitudes

• Acceleration: At equilibrium

State the formula for displacement of a particle in S.H.M (where the particle is at its maximum displacement at t=0)

x = x₀cos(ωt)

State the formula for displacement of a particle in S.H.M (where the particle is at its equilibrium position at t=0)

x = x₀sin(ωt)

State the formula for the velocity of a particle in S.H.M where x = x₀cos(ωt)

v = -ωx₀sin(ωt) = -ω√(x₀² - x²)

State the formula for acceleration of a particle in S.H.M where x = x₀cos(ωt)

v = -ω²x₀cos(ωt) = -ω²x

State the formula for the angular frequency of a mass-spring system

ω = √(k/m)

State the formula for the period of a mass-spring system

T = 2π√(m/k)

Derive the formula for angular frequency of a pendulum

• The restoring force is the tangential component of the weight, -mgsinθ

• For small angles, like θ < 10°, sinθ ≈ θ by small angle approximation

• Hence, restoring force = -mgθ = -mg(s/L)

• By N2L, -mg(s/L) = ma ⇒ a = -(g/L)s

• Comparing to a = -ω²x, ω = √(g/L)

State the formula for the period of a pendulum

T = 2π√(L/g)

State the kinetic energy of a particle in S.H.M

½mω²(x₀² - x²)

State the potential energy of a particle in S.H.M

½mω²x²

State the formula of the total energy of a particle in S.H.M

½mω²x₀²

State the formulae which shows how kinetic energy and potential energy varies with time

E_k = ½mω²x₀²sin²(ωt)

Eₚ = ½mω²x₀²cos²(ωt)

Define damping

Damping is the process whereby energy is removed from an oscillating system

State the 3 types of damping

• Light damping

• Critical damping

• Heavy damping

Describe light damping

Light damping occurs when resistive force is small. This results in oscillations where the amplitude decays exponentially with time.

Describe critical damping.

Critical damping occurs when the system experiences a resistive force stronger than that in light damping which prevents the system from oscillating. The system returns to the equilibrium position in the shortest time after its initial displacement.

Describe heavy damping.

Heavy damping occurs when the system experiences an even stronger resistive force than critical damping which prevents the system from oscillating. The system takes a longer time to return to its equilibrium position after its initial displacement (as compared to critical damping).

Define forced oscillations

They are oscillations which are produced when a body is subjected to a periodic external driving force and is made to oscillate at the frequency of the driving force, which may not be its natural frequency.

State what the driver does in forced oscillations

It provides the periodic driving force

Define resonance

It is the thing which occurs when the frequency of the driving force is close to or at the natural frequency of the driven system. There is maximum transfer of energy from the driver and the system responds with maximum amplitude.

State the 2 things the amplitude of a forced oscillation depends on

• Damping of the system

• The relative values of driving frequency and natural frequency of the driven system (the closer they are, the closer to the maximum amplitude is reached)

Describe how the sharpness of the frequency response curves changes with damping

• With no damping, resonance occurs at natural frequency

• With light damping, resonance occurs at a frequency that is close to the natural frequency, the resonance peak is sharp

• With increasing damping (still light damping), resonance occurs at lower frequency and with a smaller maximum amplitude. The resonance peak broadens.

• At and past the critical damping, the frequency response curve is relatively flat.