Question

The motion of a particle executing simple harmonic motion is described by the displacement function,

x (t) = A cos (ωt + φ).

If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s^–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

Answer 1

Initially, at t = 0:

Displacement, x = 1 cm

Initial velocity, v = ω cm/sec.

Angular frequency, ω = π rad/s^–1

It is given that:

Squaring and adding equations (i) and (ii), we get:

Dividing equation (ii) by equation (i), we get:

SHM is given as:

Putting the given values in this equation, we get:

Velocity,

Substituting the given values, we get:

Squaring and adding equations (iii) and (iv), we get:

Dividing equation (iii) by equation (iv), we get: