SIMPLE HARMONIC MOTION

 

SIMPLE  HARMONIC MOTION

Simple Harmonic Motion (SHM) is the periodic motion of a body or particle along a straight line such that the acceleration of the body is directed towards a fixed point and is also proportional to its displacement from that point.

Examples of SHM

1.        Mass suspended from spring.         II. Loaded test-tube in a liquid   III. The simple pendulum

4.        Prongs of a sounding turning fork  V. motion of balance wheel of a clock.

TERMS IN SHM

1.        Amplitude (A): of a simple harmonic motion is the maximum displacement of the body performing simple harmonic motion from its equilibrium or centre position.

2.        Period (T): is defined as the total time taken by a vibrating body to make one complete revolution (or cycle) about a reference point. T  = t/n

3.        Frequency (f) is the number of complete revolutions per second made by a vibrating body. F = 1/T.

Angular velocity (w) = angle turned through by the body / time taken

                   W = ᴓ/t (rad s-1) ; w = 2ᴧf

V = wA (Linear speed equals the product of the angular speed and the radius or amplitude of motion.)

Acceleration, a = -w2x

The linear acceleration a equals the product of the square of the angular speed and the displacement, x, of the particle from the centre of motion.

Angular acceleration (a) of a body is the time rate of change of its angular velocity (w). It is expressed in radians per second per second (rads-2)

 a = αr ( where a is the linear acceleration, α is the angular acceleration and r is the radius or the displacement of the particle from its central position).

SHM in simple pendulum 

T= 2

Energy of Simple Harmonic Motion (SHM)

Maximum velocity of simple pendulum (Vm) =

Total energy at any instant of motion of simple pendulum ET = ½ mvm2.

THEORY OF VIBRATION OF A LOADED SPRING

Maximum total energy stored in the spring is given by : W =1/2 KA2.  Where A is the amplitude of motion and K is the force constant.

Velocity of the suspended mass at a point y from the equilibrium position is given as V = (A2-y2) or V = w A2-X2

The period (T) of loaded spring , T= 2   where the true value of the period (T) is given by  T = 2  

i.e Ꙍ =

FORCE VIBRATION AND RESONANCE

A system performing S.H.M gradually loses its energy due to friction within its parts and air resistance. The amplitude of such a motion gradually becomes smaller and smaller with time until it decreases eventually to zero. Such a motion is said to damped.

Force vibration is the vibration resulting from the action of an external periodic force on an oscillating body.

Forcing Frequency: is the frequency generated when object has subjected to an external periodic force.

Resonance : is said to occur when the forcing frequency (f) of an external periodic force coincides with the natural frequency (f0) of a body with which it is in contact, causing the body to vibrate with a large amplitude.

Example

1.  A body moving with SHM has an amplitude of 10 cm and a frequency 100 Hz. Find (a) the period of oscillation (b) the acceleration at maximum displacement  (c ) the velocity at the centre of motion.

Solution

A = 10 cm = 0.1 m ; f = 100 Hz

(a) T =  =1/100  =  0.01s      (b)  a = - w2A = (2 )2A = 4 x 1002 x 0.1 x 2 = 4 x103 2 ms-2.

( c) velocity at the centre, (linear velocity) Vm = w A = 2  fA  = 2 x 100 x 0.1  = 20  m/s.

Assignment

1. if a body moving with simple harmonic motion has an angular velocity of 50 radians per second, and an amplitude of 10 cm, calculate its linear velocity.

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