Question

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: [x is in cm and f is in s].

(a) x = -2 sin(3t + π/3)

(b) x = cos (π/6 - t)

(c) x = 3 sin (2πt + π/4)

(d) x = 2 cos πt

Answer 1

(a) 

Comparing above with standard equation,

We have, A = 2 cm: ϕ = 5π/6: θ = 150° and 5π/T = 3 or w = 3 rad s^-1.

(b) the reference circle is shown in the Fig.(b) above

x = cos(π/6 - t)

Comparing it with standard equation,

x = cos[2π/T . t + ϕ]

We have, A = 1 cm

2π/T = ω = 1 rad s^-1, ϕ = -π/6

(c) The reference circle is shown in the Fig. (c) above.

x = 3 sin(2πt + π/4)

The above equation can be written as

Comparing it with x =

We get, A = 3 cm, ϕ = 3π/4

(Negative sign shows that it starts on the negative x-axis. Therefore circle is shown in Fig.(c)).

(d) For, x = 2 cos πt

Comparing it with

x = A cos (wt + ϕ)

We have, A = 2 cm, ϕ = 0, w = π

The reference circle is shown in Fig. (d).