Simple Harmonic Motion

> An object is oscillating when it has a motion that is repetitive.

EG. A simple pendulum.

> Period, T - The time taken for one complete cycle.

> T = 1 / f

> Amplitude, a - The size of the oscillation.

> Phase difference - In an oscillation of the same period, T, If bothy objects reach the maximum displacement at the same time, they are in phase. However if they arrive at different times, the difference in t between there arrivals is known as there phase difference.

> Δt / T

> To find the phase difference in radians you use the formula;

> 2π Δt / T

Experiment: 3.1 Investigating oscillations.

> Simple - Physically and mathematically simple

> Harmonic - Repeats the same pattern repeatedly

"The motion of a particle whose acceleration is directed towards a fixed point and

whose acceleration is proportional to the displacement from that point."

> In this type of motion the displacement, velocity and acceleration are constantly

changing in both magnitude and direction.

> In Simple Harmonic Motion:

> Accleration α Displacement

> Acceleration is always in the opposite direction to displacement.

Acceleration, a = - constant X displacement,x

constant = (2πf)²

Therefore, a = -(2πf)²x

Additional Information: Pg. 36 Fig.2(iii)

> Ball P

Uniform Circular Motion

> Acceleration, a = -v² / r

> Speed, v = 2πrf

> Therefore, a = - (2πrf)² r

See notes for Ball Q

Additional Information:

> x is the displacement from = r to -r

> Amplitude A = r, When, x = A.

> x = A cos θ

> Angular displacement(rads)

> θ = 2πt / T = 2πft

> Hookes Law: Materials and objects deform proportionally to the force applied within a limited range of forces.

> Tension = k X extension,e

> k - Spring constant (Nm-¹)

> When a mass, m is hung from a spring

> Tension = mg = ke

> When the mass on the spring is pulled down by a distance of x from its rest position, O the object is no longer in equilibruim.

> Stretching tension = k ( e + x )

> Restoring Force = k ( e + x ) - mg

Restoring Force = ke - kx - mg

Restoring Force = -kx ( Since mg = ke )

> When the mass is released it oscillates:

> F = m a = -k x

> a = - k x / m

> a = α - x

> a = - ω² x

> ω² = - a / x = k / m

> T = 2π / ω

> T = 2π √ m / k

Experiment: 3.4 Simple pendulum pratical

> Free Oscillations - If an object is oscillating at a constant amplitude then it has no friction acting on it. The only forces effecting the object are the ones that combine to proivde the objects restoring force. If friction was present then the amplitude was gradually descrease and the object would eventually stop osciallating.

> The Potention energy, PE, changes with displacement, x.

> Ep = 1/2 k x ^2

The total energy is therefore: > Et = 1/2 k A^2

> Since, Et = Ek + Ep

> Ek = 1/2 k ( A^2 - x^2 )

Since:

> Ek = 1/2 m v²

> 1/2 m v² = 1/2 k ( A² - x² )

Also:

> v² = (2πf)² ( A² - x² )

> v = +/- (2πf) √ A² -x²

See Pg. 45 Fig. 2

KE+PE == 1/2 k A²

> Therefore when the two lines are compared to total of there points gives you the total energy of the object.

Forced oscillations - A system undergoes a forced oscillation when a periodic force is apllied to it.

Resonance - An object is in resonance when the frequency being applied to it is equal to its natural frequency.

Applied frequency of periodic force = natural frequency of object

As the applied frequency becomes greater than the natural frequency the following effects happens:

> The amplitude of the oscillations decreases

> The phase difference increases from 1/2π to π(rad)