5.3 Oscillations

A-Level Physics (OCR A)

Displacement

Distance from equilibrium position (positive or negative)

Amplitude

The maximum displacement

Period

Time taken to complete one full oscillation

Frequency

Number of complete oscillations per unit time

Phase difference

Difference in displacement between 2 oscillating objects or the difference in displacement of 1 oscillating object at different times

Antiphase

2 oscillating objects. 1 is at maximum positive displacement at the same time that the other one is at maximum negative displacement. Phase difference = 𝛑 rad

In phase

2 oscillating objects reach maximum positive or negative displacement at the same time. Phase difference = 0 rad

Angular frequency definition

Used to describe the motion of an oscillating object. Closely related to the velocity of an object in circular motion.

Angular frequency equations

ω=(2𝛑)/T or ω=2𝛑f

Simple harmonic motion definition

Oscillating motion. ω² is constant. Acceleration (a) is directly proportional to displacement (x). Acceleration acts in opposite direction to displacement (to the equilibrium point)

Simple harmonic equation

a=-ω²x

The rules of simple harmonic motion

1. Simple harmonic oscillator exhibit sinusoidal motion

2. Displacement is proportional to negative acceleration (because acceleration is always towards the equilibrium centre)

3. Period is isochronous for all amplitudes (takes the same amount of time!)

$$x=Asin\omega t$$

Displacement equation

$$v=A\omega cos\omega t$$

Velocity equation

$$a=-A\omega^2\sin\omega t$$

Acceleration equation

$$T=2\pi\sqrt{\frac{m}{k}}$$

Period equation

Force constant of springs in series

(Force constant of one spring)/(number of springs)

Force constant of springs in parallel

(Force constant of one spring)*(number of springs)

Derive the period equation ($T=2\pi\sqrt{\frac{m}{k\left(system\right)}}$ )

F=-kx and F=ma so -kx=ma

Rearrange to get a=-kx/m

The simple harmonic motion equation is a=-w²x

So -kx/m=-w²x

Rearrange to get w=(square root k/m)

w=2(pi)f so (square root k/m)=2(pi)f

so T=2(pi)(square root m/k)

Net force equation for double spring simple harmonic motion

Fnet=-(k1+k2)s

With respect to time, elastic potential energy equation (horizontal)

$$\frac12kA^2\sin^2\omega t$$

With respect to time, kinetic energy equation (horizontal)

$$\frac12mA^2\omega^2cos^2\omega t$$

$$\frac12mv^2=\frac12k(A^2-x^2)$$

Describing energy with respect to time equation (horizontal)

With respect to time, which forms of energy are there in a system? (Vertical)

GPE, KE, EPE

Is velocity proportional to amplitude?

Yes

Is energy proportional to amplitude squared?

Yes

What is damping?

When an external force acts on an oscillating object and reduces the amplitude of the oscillations

What is light damping?

When damping forces are small so changes over time are small

What is heavy damping?

When damping forces are large so large changes happen over a small amount of time. Sometimes with heavy damping there can be no oscillatory motion

What are free oscillations?

Oscillations from an oscillator given a displacement and then let go. The frequency of oscillations is defined as “natural frequency”

What are forced oscillations?

Oscillations from an oscillator driven by external forces. The frequency is defined as “driving frequency”

What happens if driving frequency and natural frequency is the same in a system?

The object will resonate, which increases the amplitude of oscillations dramatically! If the system is not damped then it may break.

What does a graph of amplitude vs driving frequency look like? Include where the natural frequency is.

Take care to note that it is asymmetrical!!!

What does a graph of amplitude vs driving frequency look like when damped? Include where the natural frequency is.