Vertical Motion of a Spring-Mass System

Prove that a mass attached to the end of an ideal vertical spring undergoes SHM when displaced from the equilibrium position.

The Simple Pendulum

Prove that, for small angles, the motion of a pendulum approximates SHM.

• A simple pendulum also exhibits periodic motion

• The motion occurs in the vertical plane and is driven by gravitational force

Mathematical Description of Simple Harmonic Oscillation

Basic equation of motion for SHM

It is often convenient to write this as:

This is a differential equation:

Proof 4:

Prove that general solution for the equation

has a form:

Constants A and B depend on the initial conditions i.e. value of x and v when t = 0.

The another possible form is

Graph of x(t) = A sin(ωt)

Initial conditions at t = 0 are:

x (0) = 0

v (0) = v_max

The graphs of velocity and acceleration are shifted one-quarter cycle to the right compared to the graph above it, where x(0) = 0

Graph of x(t)=A cos(ωt)

Initial conditions at t=0 :

x (0) = A

v (0) = 0

It is similar to expression

x(t)=A sin (ωt+π/2)

with initial phase of π/2

Physical meaning of this solution

At t = 0 then x = 0, so the motion “starts” when the mass is passing through the equilibrium position

x varies sinusoidally and the motion repeats with period

and frequency

ω is the angular frequency

A is called the amplitude and is the maximum displacement.

The maximum speed is v_max = Aω

The maximum acceleration is a_max = Aω²

For a spring

For a simple pendulum

Velocity V and acceleration a in terms of displacement

Acceleration

Velocity

Find v² in terms of displacement, not time

Energy in SHM

The energy required to extend the spring by a displacement x is stored in the spring and it is called spring potential energy

Hence the total SHM energy is given by:

Maximum KE is

Maximum PE is

Proof 1: show that

Damped oscillations

SHM is more realistic if it includes friction, air resistance, etc. (damping). The equation of motion becomes:

Proof 2: show that the total rate of energy loss is equal to the power dissipated through friction:

Overdamping at b>

Underdamping at b<

Periodic Driving Force

Apply a periodic force to a damped system undergoing SHM at its natural frequency.

Consider a periodic force

And the equation of motion becomes:

Resonance

• Resonance (highest amplitude peak) occurs when the driving frequency, ω, equals the natural frequency, ω₀;

• Amplitude increases as damping decreases;

• The amplitude vs. frequency curve (resonance curve) broadens as damping increases;

• The shape of the resonance curve depends on the damping coefficient b.

Introduction to waves

Longitudinal waves

In a longitudinal wave, the direction of vibration of each particle is parallel to the wave’s direction of motion.

The sound wave travels horizontally, as individual particles oscillate back and forth

Transverse waves

In a transverse wave, the direction of vibration of each particle (or field) is perpendicular to the wave’s direction of motion.

Case Study: Surface-water waves

Superposition of longitudinal and transverse waves. Water elements move in circles, but the wave travels horizontally

Types of waves

A mechanical wave requires a propagation medium, and it can be a transverse, longitudinal, progressive or standing wave;

EM waves do not need a propagation medium. EM waves are always transverse, but they could also be stationary or progressive.

Longitudinal Waves: Sound Waves, Seismic p-waves

Transverse Waves: Mechanical Waves on strings, Electromagnetic Waves

The Sinusoidal Travelling Wave Model

- Crest: the point of maximum value or upward displacement;

- Trough: the minimum-value or lowest point;

- Amplitude: the distance between a crest or trough and the equilibrium position (zero displacement);

- Wavelength (λ): the distance between adjacent crests or troughs;

- Period (T): the time it takes a wave to travel a distance equal to λ.

• The simplest wave is sinusoidal;

• Let x be the distance measured in the direction of travel and y the particle displacement;

• Suppose that y = 0 at x = 0

An equation representing such a wave at some instant of time is:

The equation of a moving sine wave

The wave travels a distance x = λ in a time t = T Then:

Waves and SHM

In general, every point on the wave undergoes Simple Harmonic Motion (SHM);

For any point with x ≠ 0 the displacement y is a harmonic function of both time and distance from the origin;

At a particular x as x can be treated as a constant. This is the SHM equation;

NOTE: Although the wave travels at constant speed, each point in the wave is in SHM. The SHM period is the same as that of the wave.

A mechanical wave:

• when a travelling, wave transports and transfers energy not matter;

• is produced by a local oscillation, perturbation or disturbance in the material;

• requires a medium (e.g. air, water, etc.) to propagate;

• can be described by Newton’s laws of motion, hence the name “mechanical waves”.

The propagation speed

- A longitudinal sound wave consists of alternating high (compressions) and low pressure (rarefactions) regions. Ears detect the pressure differences directly;

- The phase difference between particle displacements and pressure or density is 90⁰;

- The speed of sound increases with temperature and mass density of material;

- Audible Sound: (20 Hz < f < 20 kHz).

Energy and Intensity of sound waves