Simple Harmonic Motion


Simple Harmonic Motion (SHM) is defined as the oscillatory motion of a particle whose acceleration a is always directed towards a fixed point and is directly proportional to its displacement x from that fixed point but in the opposite direction to the displacement.

Negative sign indicated that the acceleration is always in a direction opposite to the displacement.

Fixed point is usually called the equilibrium position, the position at which no net force acts on the oscillating particle.

When the particle is displaced from the equilibrium position, it experiences a restoring force which tends to bring it back to the equilibrium position.

  • Restoring force always directed towards equilibrium position

 

 

Velocity of SHM:

$v = \omega x_{o} cos(\omega t)$ or $v = – \omega x_{o} sin (\omega t)$

$v = \pm \omega \sqrt{x_{o}^{2} – x^{2}}$

At equilibrium (x=0) position, $v_{min} = \pm \omega x_{o}$

At position of maximum displacement ($x = x_{o}$), $v_{min} = 0$

 

Acceleration of SHM:

 $a \propto -x$ $\rightarrow$ $a = – \omega^{2} x$

$a = – \omega^{2} x_{o} \, cos (\omega t)$ OR $a = \omega^{2} x_{o} \, sin (\omega t)$

The acceleration of the particle in SHM is not constant as it varies proportionally with displacement. Hence, kinematics equations of motion cannot be applied.

 

Kinetic Energy:

$E_{k} = \frac{1}{2} m \omega^{2} x_{o}^{2} cos^{2} \, (\omega t)$ OR

$E_{k} = \frac{1}{2} m \omega^{2} (x_{o}^{2} – x^{2})$

 

Potential Energy:

$E_{p} = \frac{1}{2} m \omega^{2} x_{o}^{2} sin^{2} \, (\omega t)$ OR

$E_{p} = \frac{1}{2} m \omega^{2} x^{2} $

 

Total Energy:

$E_{T} = \frac{1}{2} m \omega^{2} x_{o}^{2}$


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