Force on a body in simple harmonic
motion
The body is in simple harmonic
motion because there is a force acting towards the mean position. Generally
this force could be called as a restoring force. As we have derived a equation
for acceleration we can derive the equation for the force using the Newton’s
second law of motion as shown below. The constant is called as force
constant. Basing on the derivation we can again show that the acceleration is
directly proportional to displacement but in the opposite direction.
Potential energy of your body in simple harmonic motion
We can derive the equation for
the potential energy of a body in simple harmonic motion using the basic
concept of integration. Let us consider a particle having a displacement from the
mean position. To displace the body by a small value we shall do some small
amount of the work. The total displacement is some of this kind of so many
displacements therefore we have to calculate the work and for each part and add
all of them to get the total work done in this process.A mathematical phenomenon called integration has to be used to do this job.
The equation that we have derived is actually equation for the displacement of body by a smaller value. Any way
after the work is done as per the law of conservation of energy, the work can not disappear.
Work and energy are simialar kind the terms like you’re having energy, then
you can do the work and vice versa.
Therefore after doing the work, this work that will be stored in the format of energy and that energy is called potential energy of the body in simple harmonic motion. We can derive the equation for the potential energy basing on the concept of integration as shown below.
Therefore after doing the work, this work that will be stored in the format of energy and that energy is called potential energy of the body in simple harmonic motion. We can derive the equation for the potential energy basing on the concept of integration as shown below.
It could be easily noticed that
potential energy is maximum at the extreme position and intreme position
and it is minimum at the mean position.
Kinetic energy of your body in simple harmonic motion
As the body in simple harmonic
motion is having some velocity that will automatically process kinetic energy
also. The derivaetion for the kinetic energy is comparatively quite simple as
shown below. It could be easily noticed that kinetic energies is maximum at the
mean position and as it is going from mean position to extreme position it will
be keep on decreasing and finally becomes zero.
We shall notice that at one point
if the potential energies maximum at the same point kinetic energies minimum
and vice versa. This clearly satisfy the conservation of energy that the total sum of the energy always remains constant. If one energy
increases another energy decreases, but the total energy of the system always
remains constant. Energy is neither created nor destroyed it just converts from
one format to another format. This is called law conservation of energy which
is valid for the entire universe in all conditions.
In the following diagram we have
shown how to derive the question for kinetic energy and the total energy.
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