Simple harmonic motion is a simple vibratory
motion in which any body continues to move to and fro about its mean position.
For example: motion of pendulum, motion of perpendicularly hanged body with
spring.
Explanation: Consider the motion of a mass attached with a spring. Its one end is
attached to a firm support and a mass “m” is attached to its other end.
Currently the spring is in equilibrium position. If an external force “Fext”is
applied on the mass “m”, length of the spring increases by an amount “x”. The external
force “Fext” acting on the body is directly proportional to the
increase in length “x”.
This means that
the acceleration of the body is directly proportional to its displacement and
is always directed towards its mean position. As the mass “m” moves towards the
equilibrium position its displacement goes on decreasing. Resultantly the
acceleration of the body also decreases. On reaching the equilibrium position,
displacement becomes zero also acceleration becomes zero. But at this point
velocity is maximum, so due to inertia the mass does not stop at this position
but continues its motion towards left and the spring is being compressed. Hence
the body of mass “m” keeps on vibrating.
RESTORING FORCE.
When an external
force is applied on a spring, the length of the spring increases. After
removing the force the spring moves towards its equilibrium position. The force
due to which it moves towards its equilibrium position is known as restoring
force.
SIMPLE PENDULUM & ITS MOVEMENT
Definition:
A simple pendulum consists of a single isolated bob suspended from a
frictionless support by means of a light inextensible string.
Explanation:
The motion of simple pendulum is also simple harmonic motion. In equilibrium,
the pendulum is held stationary in a vertical position. If the bob is disturbed
from its equilibrium position, it will start moving towards its mean position
under the action of one component of gravitational force. At equilibrium
position the velocity of the bob is maximum and due to inertia it passes by its
mean position towards the other end. But now the velocity of the bob decreases,
and becomes zero as it reaches at its maximum point on the other end. And then
bob starts moving between one end to the other end. The acceleration of the bob
always remains toward the mean position.
CHARACTERISTICS OF SIMPLE HARMONIC MOTION
(i) A body executing simple harmonic motion
always vibrates about its equilibrium position.
(ii) Its acceleration is always directed
towards its mean position.
(iii) Its
acceleration is directly proportional to its displacement from the mean
position. i.e 0 at mean position and maximum at extreme position.
(iv) Its velocity is maximum at the mean
position and zero on the extreme position.
Mathematical Problems
Problem 1: The
time period of a simple pendulum is 2s. What will be its length on earth and on
moon?
Problem 2: Length
of a spring is 8 cm. when a mass of 4 Kg is hung to it, its length becomes 16
cm. what is its spring constant?
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