Lecture-8: Chapter-2 (Waves & Oscillation)

Simple harmonic motion 

Periodic motion: A motion which repeats along the same path after a definite interval of time is called periodic motion. Earth’s motion around the Sun, motion of hand or wall clock, and motions of planets are example of periodic motion.

Oscillation/ vibration: When a body moves back and forth repeatedly about a mean position, its motion is called oscillation or vibration. Motion of a simple pendulum, vibration of a tuning fork, motion of a spring etc are examples of oscillation or vibration.

Simple harmonic oscillation: The type of vibratory motion of a body such that the restoring force or the acceleration acting on the body is directly proportional to the displacement from the mean position and always directed towards the mean position is called the simple harmonic oscillation or motion. Motions of a simple pendulum with small amplitude, vibration of arms of tuning fork, motion of a spring etc are examples of simple harmonic oscillation or motion (SHM).

Characteristics of simple harmonic oscillation: A simple harmonic motion has the following characteristics:

1. Its motion is periodic, 2. It particular time interval the motion becomes opposite, 3. Its motion is along a straight line, 4. Its acceleration is proportional to the displacement, 5. Acceleration is opposite to displacement and 6. Acceleration points towards the mean position of the object.

Expression of displacement, velocity and acceleration of a particle executing Simple harmonic motion:

a) Displacement:

Let a particle move around a point O in a circular path ABCD of radius A at an angular velocity ω in the direction shown by the arrow in fig-3. Let at time t it comes to point P from point A. From P a normal is drawn on the diameter DB. Here the displacement of the end point of the normal is, x = ON.

Now, from fig-3, , here θ is the angular displacement.

We, get, ON/OP = sin θ  or, ON = OP X sin θ  or, x = Asin θ

Or, x = Asinωt……………(1). If T is the time period of oscillation, then ω = 2π/T = 2 πf.

So, x = Asin 2 πft………….(2). Eq. (1) and (2) are the eqs. Of displacement of a particle executing SHM.

b) Velocity: We know that the rate of change of displacement is called velocity. It is denoted by v.

Relation between velocity and displacement: We know, x = Asinωt or, sinωt = x/A and cosωt = √(1-sin²ωt).So, v = Aω cosωt = Aω √(1-sin²ωt) =

(i) When x = A, then v = 0 and (ii) when x = 0, then v = Aω

so, at the mid-point of the motion of N, the velocity will be maximum and with the increase of displacement velocity will start decreasing. At the extreme ends, i.e., at B or D its velocity will be zero.

c) Acceleration: We know the rate of change of velocity is called acceleration. It is denoted by a.

This is (above) the relation between acceleration and displacement. Negative sign indicates that acceleration and displacement are opposite to each other.

(i) when x = 0, then a = 0 and (ii) when x = A, then a= -ω²A. That is at maximum displacement of the motion of N, acceleration is maximum and at mid point it is zero.

Differential equation of simple harmonic oscillation

Let a particle of mass m oscillate in simple harmonic motion. Now, at time t if its displacement is x, then

Velocity, and acceleration, . Now, magnitude of force acting on the particle,

Since force or acceleration is proportional to displacement and is in opposite direction, so

………….(1)  where K is a constant, called force constant.

Again, if the angular velocity of the particle is ω, then we get, ………………..(2)  [By eq. (1)]

Comparing eq. (1) and (2), we have ……… (3). This is the differential equation of a particle executing SHM.

Solution of differential equation of SHO: The differential equation of a particle executing SHM is given by

………….(1). To solve eq. (1), let us multiply both sides by , then

….(2). Integrating eq. (2), we get, ……….(3), where c is the integration constant. We need to find out the value of it.

When, x = A, then dx/dt = 0. Inserting this condition in eq. (3), we get, c = ω²A². So,

…..(4). Integrating eq.(4), we get , here  is the integration constant. ………(5). This is the general solution of differential equation of SHM or, SHO.

Prob-1: The amplitude and frequency of an object executing SHM are 0.01 m and 12 Hz, respectively. What is the velocity of the object at displacement 0.005 m? What is the maximum velocity of the object?

Solution: a) We know, =0.653m/s.

b) We know, when x = 0, then v = v_max. so,

Prob-2: The time period of an object executing SHM is 0.001s and its amplitude is 0.005m. Calculate its acceleration at 0.002m from the mid-point of the motion.

Solution: a) We know, a = ω²x =

Progressive waves

Definition: If the wave generated from a source progresses with time from one point to another through a medium, it is called progressive wave. Progressive wave can be longitudinal or transverse.

Characteristics of Progressive waves:

1. These waves are generated by continuous disturbance of a portion of a medium.
2. These waves travel with a fixed velocity through a uniform medium.
3. The velocity of propagation of the waves depends on the density and elasticity of the medium.
4. Vibrations of the particles of medium may be transverse or longitudinal.
5. As the wave progresses, every point of the medium undergoes same change of pressure and density.

Equation of progressive waves:   or,

Stationary Waves

Definition: The resultant wave produced by the superposition of two progressive waves, having same wavelength and amplitude, traveling in opposite directions is called stationary wave. The stationary wave has no forward motion but remains fixed in peace.

In stationary waves there are certain points where the amplitude is zero.  These points are nodes and there are some points where the amplitude is maximum. These points are called antinodes. In fig-1 points A are antinodes and N is nodes.