Derivation of Velocity and acceleration in SHM

Simple Harmonic Motion (SHM) is a type of oscillatory motion in which the displacement of a particle from its equilibrium position is proportional to the force acting on it, but in the opposite direction. The motion is periodic and the frequency of oscillation is determined by the restoring force and the mass of the particle.

Velocity in SHM

The velocity of a particle in SHM can be derived by taking the time derivative of its displacement function. Let x(t) be the displacement of the particle from its equilibrium position at time t. Then the velocity v(t) of the particle can be given by:

v(t) = dx/dt

where dx/dt is the time derivative of x(t). To obtain an expression for v(t), we need to find an equation for x(t) first.

Suppose the particle has an angular frequency ω and an amplitude A. Then its displacement x(t) can be given by:

x(t) = A cos(ωt + φ)

where φ is the initial phase angle. Taking the time derivative of x(t), we get:

dx/dt = -ωA sin(ωt + φ)

Thus, the velocity v(t) of the particle can be expressed as:

v(t) = -ωA sin(ωt + φ)

This shows that the velocity of the particle is also periodic and sinusoidal in nature, with a maximum value of ±ωA and a minimum value of zero.

Acceleration in SHM

The acceleration of a particle in SHM can be derived by taking the time derivative of its velocity function. Let v(t) be the velocity of the particle at time t. Then the acceleration a(t) of the particle can be given by:

a(t) = dv/dt

where dv/dt is the time derivative of v(t). To obtain an expression for a(t), we need to find an equation for v(t) first.

From the previous derivation, we know that the velocity v(t) of the particle can be expressed as:

v(t) = -ωA sin(ωt + φ)

Taking the time derivative of v(t), we get:

dv/dt = -ωA cos(ωt + φ)

Thus, the acceleration a(t) of the particle can be expressed as:

a(t) = -ω^2 A cos(ωt + φ)

This shows that the acceleration of the particle is also periodic and sinusoidal in nature, with a maximum value of ±ω^2 A and a minimum value of zero. The negative sign indicates that the acceleration is always directed towards the equilibrium position, which is the opposite direction to the displacement.