Chapter 10: Simple Harmonic Motion & Waves

Definition: Simple harmonic motion (SHM) is a to and fro oscillatory motion in which acceleration of the body is directly proportional to the displacement of the body from the mean position and is always directed towards the mean position.

a \propto -x

Negative sign indicates that acceleration and displacement are opposite in direction.

Motion of Mass attached to a Spring

Consider a mass 'm' attached to the end of an elastic spring. The other end of the spring is fixed at the firm support. The whole system is placed on a smooth horizontal surface.

Restoring Force: A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position. According to Hooke's Law:

F = -kx

Where k is Spring Constant. The value of k is a measure of the stiffness of the spring. Stiff springs have large k values, and soft springs have small k values.

Displacement of mass m: If we release mass 'm' at point 'A', it moves forward to 'O'. At point 'O' it will not stop but moves forward towards point 'B' due to inertia and covers the same displacement -x. At point 'B' once again elastic restoring force F acts upon it but now in the right side. In this way it continues its motion from A to B and then B to A.

According to Newton's 2nd law of motion (F=ma):

ma = -kx

a = -\frac{k}{m} x

Since k/m is constant, therefore acceleration is directly proportional to displacement a ∝ -x. Thus the motion of a mass attached to a spring is SHM.

Time Period: The time period T of the simple harmonic motion of a mass m attached to a spring is given by:

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

The motion of a ball placed in a bowl is an example of simple harmonic motion.

• At mean position: When the ball is at the mean position O, that is, at the centre of the bowl, net force acting on the ball is zero. Weight of the ball acts downward and is equal to the upward normal force of the surface of the bowl. Hence there is no motion.
• At extreme position: When a ball is gently displaced from the centre of a bowl it starts oscillating about the centre due to force of gravity, which acts as a restoring force.
• Motion: If we bring the ball to position A and release it, the ball will start moving towards O due to restoring force (component of weight). At O it attains maximum speed and moves towards B due to inertia. At B, its speed becomes zero and it moves back towards O. This to and fro motion continues until energy is lost to friction.

A simple pendulum also exhibits SHM. It consists of a small bob of mass m suspended from a light string of length L fixed at its upper end.

Forces acting on the bob: In the equilibrium position O, the net force is zero. At extreme position A, the weight mg can be resolved into two components:

• mg cos θ: along the string, balances the tension T.
• mg sin θ: perpendicular to the string, acts as the restoring force directed towards mean position.

Motion: Due to restoring force mg sin θ, the bob moves towards mean position O. At O, velocity is maximum, and it continues to B due to inertia. The acceleration is always directed towards the mean position O, hence it is SHM.

Time Period: The time period of a simple pendulum is given by:

T = 2\pi \sqrt{\frac{L}{g}}

The time period depends upon the length of the pendulum and is independent of the mass and amplitude.

Important Features of SHM

• A body executing SHM always vibrates about a fixed position.
• Its acceleration is always directed towards the mean position.
• The magnitude of acceleration is always directly proportional to its displacement from the mean position (zero at mean, maximum at extreme).
• Its velocity is maximum at the mean position and zero on the extreme positions.

Definitions

Vibration: One complete round trip of a vibrating body about its mean position is called one vibration.

Time Period (T): The time taken by a vibrating body to complete one vibration is called time period.

Frequency (f): The number of vibrations per cycle of a vibrating body in one second is called its frequency. It is reciprocal of time period i.e., f = 1/T.

Amplitude (A): The maximum displacement of a vibrating body on either side from its mean position is called its amplitude.

Definition: The oscillations of a system in the presence of some resistive force are damped oscillations.

Explanation: Friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped. This damping progressively reduces the amplitude of the motion.

Application: Shock absorbers in automobiles are a practical application. They damp violently vibrations caused by road bumps and convert energy into heat energy of the oil.

Definition: A wave is a disturbance in the medium which causes the particles of the medium to undergo vibratory motion about their mean position in equal intervals of time.

Waves as Carriers of Energy: Energy can be transferred from one place to another through waves. For example, shaking a string transfers muscular energy to the other end. Water waves transfer energy from the stone's impact point to the shore. Matter itself is not transferred.

Categories of Waves

• Mechanical Waves: Waves which require any medium for their propagation are called mechanical waves. Examples: water waves, sound waves, waves on strings and springs.
• Electromagnetic Waves: Waves which do not require any medium for their propagation are called electromagnetic waves. Examples: Radio waves, television waves, X-rays, heat and light waves.

Mechanical waves may be classified as longitudinal or transverse.

Longitudinal Waves

In longitudinal waves the particles of the medium move back and forth along the direction of propagation of wave.

• Production: Can be produced on a slinky (spring). Push and pull one end of the slinky along its length.
• Compression: Regions where loops of spring are close together.
• Rarefaction: Regions where loops are spaced apart.

Transverse Waves

In the case of transverse waves, the motion of particles of the medium is perpendicular to the motion of wave.

• Production: Move the slinky up and down.
• Crests: The highest points of the particles of the medium from the mean position.
• Troughs: The lowest points.

The relation between the Velocity, frequency and wavelength of the wave is known as wave equation.

v = f\lambda

Where: v = speed of wave, f = frequency, λ = wavelength.

Definition: Ripple tank is a device to produce water waves and to study their characteristics.

Construction: A rectangular tray with a glass bottom placed above a table. A vibrator (paddle) touches the water surface to generate waves. An electric bulb hangs above to project the wave pattern (crests as bright lines, troughs as dark lines) onto a screen.

Properties of Waves studied with Ripple Tank

1. Reflection: When waves moving in one medium fall on the surface of another medium, they bounce back into the first medium such that the angle of incidence is equal to the angle of reflection.
2. Refraction: When waves from one medium enter in the second medium at some angle their direction of travel may change. Speed of water waves depends on depth (slower in shallow water). When entering shallow water, wavelength decreases but frequency remains same.
3. Diffraction: The bending or spreading of waves around the sharp edges or corners of obstacles or slits is called diffraction. Diffraction is clearer if the size of the obstacle/slit is comparable to the wavelength.