Subject: Physics
As the restoring force acts in the S.H.M., the particle executing S.H.M has both P.E. and K.E.
Potential Energy
Suppose a particle of mass m is executing S.H.M . with amplitude r and angular velocity . If y is the displacement, then acceleration
\begin{align*} a &= -\omega ^2y \\ \text {restoring force,} F &= ma \\ &= -m\omega ^2 y =-ky \end{align*}
Where \(m\omega ^2 = k,\) a constant. If a particle is displaced further by a small displacement dy against the force, then work done
$$dW = -Fdy =-(-ky) dy = kydy $$
Therefore total work done to displace the particle from mean position to the position of displacement y is
\begin{align*} W &= \int _0^y Fdy = \int _0 ^y ky dy = k \left [ \frac {y^2}{2}\right ]_0 ^y = \frac 12 ky^2 \\ \therefore W &= \frac12 ky^2 \end{align*}
The work done on the particle will remain in the form of potential energy. Thus,
\begin{align*} E_p &= \frac12 ky^2 \\ &= \frac 12 \omega ^2 y^2 \dots (i) \end{align*}
Kinetic Energy
Kinetic energy of the particle with velocity v is given by
\begin{align*} E_k &= \frac12 mv^2 \\ &= \frac 12 m(\omega \sqrt {r^2 - y^2})^2\\ &=\frac 12 m(\omega ^2 r^2 - y^2) \dots (ii) \end{align*}
Total energy of the particle at any point is
\begin{align*} E &= E_p + E_k \\ &= \frac 12 \omega ^2 y^2 + \frac 12 \omega ^2 (r^2 - y^2) \\ &=\frac 12 \omega ^2 r^2 \\ \therefore E &= \frac 12 \omega ^2 r^2 \\ \text {or,}\: E &= 2m\pi ^2 f^2r^2 \dots (iii) \end{align*}
As m, v and r are constants, the total energy remains constant for a particle executing S.H.M.
When force such as friction, viscous force acts on a body executing S.H.M, the forces offer resistance to the motion. As a result, the mechanical energy of a body gradually decreases. The vibration of body in motion also decreases and finally comes to rest. This type of motion is called damped oscillation. Example: motion of a pendulum in a liquid, to and fro motion of a metallic strip in a metallic strip in a magnetic field.
When a system oscillating is given some initial displacement from its equilibrium position and left free, it begins to oscillate with its own frequency with constant amplitude. Then the oscillation of the body is called free oscillation. The natural frequency of vibrations of free oscillating system depends upon inertia, elastic properties and dimension of the object. Examples: pendulum, tuning fork or string in vacuum.
Forced Vibration and Resonant Vibration
If a body is set in vibration by an external periodic force whose frequency is equal to the natural frequency of the vibrating body, the amplitude of vibration increases at each step and becomes very large. Such vibration is called resonant vibration.
Figure shows the vibration of amplitude with frequency for different values of damping. From these curves it is clear that smaller the damping taller and narrower the resonance peak. When the natural frequency of vibration is equal to frequency of applied force, the amplitude becomes maximum. This condition is called resonance. It is a special case of forced vibration.
Total work done to displace the particle from mean position to the position of displacement y is
\begin{align*} W &= \int _0^y Fdy = \int _0 ^y ky dy = k \left [ \frac {y^2}{2}\right ]_0 ^y = \frac 12 ky^2 \\ \therefore W &= \frac12 ky^2 \end{align*}
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