Simple Harmonic Motion – SS2 Physics Lesson Note (First Term)

Simple harmonic motion (SHM) is the periodic motion of a body or particle along a straight line such that its acceleration is directly proportional to its displacement from a fixed point and is always directed towards that point. The particle moves to and fro under the influence of a restoring force, which directs it back to its equilibrium position.

Behavioural objectives

By the end of the lesson, students should be able to:

• define simple harmonic motion and give examples;
• define angular velocity and relate linear velocity to it using v = rω;
• state where the velocity of a body in SHM is maximum and where it is minimum;
• show that the acceleration in SHM is a = -ω²x;
• calculate the period, frequency, total energy, maximum velocity and maximum acceleration of a mass on a spring;
• explain forced vibration and resonance.

What are examples of simple harmonic motion?

1. A loaded test tube in a liquid
2. A mass on a spring
3. The simple pendulum

How are angular velocity and linear velocity related in SHM?

As the particle P moves around the circle once, it sweeps through an angle θ = 360° (or 2π radians) in the time T, the period of the motion. The rate of change of the angle θ with time (t) is known as the angular velocity, ω.

Angular velocity (ω) is defined as the angle turned through by the body divided by the time taken.

ω = θ/t (rad/s)

θ = ωt

This is similar to the relation distance = uniform velocity × time (s = vt) for motion in a straight line.

As the angle changes with time, the arc length s also changes with time. By definition, θ in radians = s/r, hence s = rθ, where r is the radius of the circle (r = A, the amplitude).

Dividing by t gives s/t = rθ/t. Since s/t is the linear velocity v and θ/t = ω:

v = rω

Velocity at any point in SHM

The linear velocity v at any point Q, whose distance from the central point C is x, is given by:

v = ω√(A² – x²)

The maximum velocity, v_m, corresponds to the point x = 0, which is the central point or centre of motion. Hence:

v_m = ωA

Thus, the maximum velocity of the SHM occurs at the centre of the motion (x = 0), while the minimum velocity occurs at the extreme position of motion (x = A).

What is the relationship between acceleration and displacement in SHM?

x = A cos θ, and θ = ωt, so:

x = A cos ωt

dx/dt = -ωA sin ωt

dv/dt = -ω²A cos ωt = -ω²x

The negative sign shows that the acceleration is always directed inwards towards C, while the displacement is measured outwards from C.

What is the energy of simple harmonic motion?

Since force and displacement are involved, it follows that work and energy are involved in simple harmonic motion.

At any instant of the motion, the system may contain some energy as kinetic energy (KE) or potential energy (PE). The total energy (KE + PE) for a body performing SHM is always conserved, although it changes form between PE and KE.

At h = 0, PE = 0 and KE = ½mv², so KE is maximum.

When a mass is suspended from the end of a vertical spring, pulled downwards and released, it oscillates in simple harmonic motion. The force tending to restore the spring to its equilibrium position is the elastic restoring force, given by:

F = -ky

where k is the force constant of the spring and y is the displacement.

Worked example: a mass on a spiral spring

A body of mass 20 g is suspended from the end of a spiral spring whose force constant is 0.4 N m^-1. The body is set into simple harmonic motion with an amplitude of 0.2 m. Calculate:

1. The period of the motion
2. The frequency of the motion
3. The angular speed
4. The total energy
5. The maximum velocity of the motion
6. The maximum acceleration

Solution

1. T = 2π√(m/k) = 2π√(0.02/0.4) = 0.447π s = 1.41 s
2. f = 1/T = 1/1.41 = 0.71 Hz
3. ω = 2πf = 2π × 0.71 = 4.46 rad s^-1
4. Total energy = ½kA² = ½ (0.4)(0.2)² = 0.008 J
5. At the centre, all the energy is kinetic, so ½mv_m² = ½kA².
   v_m² = (0.008 × 2)/0.02 = 0.8, so v_m = 0.89 m/s.
   Or v_m = ωA = 4.46 × 0.2 = 0.89 m/s.
6. Maximum acceleration = ω²A = (4.46)² × 0.2 = 3.98 m/s²

What are forced vibration and resonance?

Vibrations resulting from the action of an external periodic force on an oscillating body are called forced vibrations. Every vibrating object has a natural frequency (f_o) of vibration. This is the frequency at which the object oscillates when it is left undisturbed after being set into vibration. The sounding board of a piano and the diaphragm of a loudspeaker work on the principle of forced vibrations.

Whenever the frequency of the external periodic force acting on a system coincides with the natural frequency of the system, the system is set into vibration with a relatively large amplitude. This phenomenon is called resonance.

Evaluation

1. Define simple harmonic motion.
2. State the expression for the velocity of a particle in SHM at a distance x from the centre. Where is the velocity maximum?
3. What does the negative sign in a = -ω²x mean?
4. What is the natural frequency of a vibrating object?
5. Define resonance.

Assignment

1. Starting from x = A cos ωt, show that the acceleration of a body in SHM is -ω²x.
2. Give three examples of simple harmonic motion.
3. Explain forced vibration and resonance, and state one use of forced vibration.