SIMPLE HARMONIC MOTION
A body is said to be vibrating if it moves back and forth or to
and fro about a point. Another term for vibration is
oscillation. A special kind of vibratory or oscillatory motion is
called the simple harmonic motion
MOTION OF MASS ATTACHED TO A SPRING
One of the simplest types of oscillatory motion is that of
horizontal mass-spring system . If the spring is
stretched or compressed through a small displacement x
from its mean position, it exerts a force F on the mass.
According to Hooke’s law this force is directly proportional to
the change in length x of the spring.
F = - k x
where x is the displacement of the mass from its mean
position O, and k is a constant called the spring constant
defined as
k = - F
The value of k is a measure of the stiffness of the spring. Stiff
springs have large value of k and soft springs have small value
of k.
F = ma
k = - ma/x
a = - x k/m
a ✔-x
It means that the acceleration of a mass attached to a spring
is directly proportional to its displacement from the mean
position. Hence, the horizontal motion of a mass-spring
system is an example of simple harmonic motion
The negative sign means that the force exerted by
the spring is always directed opposite to the displacement of
the mass. Because the spring force always acts towards the
mean position, it is sometimes called a restoring force.
A restoring force always pushes or pulls the object performing
oscillatory motion towards the mean position.
This process is repeated, and the mass continues to oscillate
back and forth about the mean position O. Such motion of a
mass attached to a spring on a horizontal frictionless surface
is known as Simple Harmonic Motion (SHM).
The time period T of the simple harmonic motion of a mass
‘m’ attached to a spring is given by the following equation:
T =2π✔m/k
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