#### 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 *t *is in *s*).

(a) *x *= –2 sin (3*t *+ π/3)

(b) *x *= cos (π/6 – *t*)

(c) *x *= 3 sin (2π*t *+ π/4)

(d) *x *= 2 cos π*t*

#### Solution 1

a) ` x = -2 sin(3t + pi/3) = + 2 cos (3t + pi/3 + pi/2)`

`= 2 cos (3t + 5pi/6)`

If this equation is compared with the standard SHM equation `x = Acos (2pi/T t + phi)` then we get:

Amplitude, A = 2cm

Phase angle, `phi = (5pi)/6 = 150^@`

Angular velocity, `omega = 2pi/T = 3 "rad/sec"`

The motion of the particle can be plotted as shown in the following figure.

If this equation is compared with the standard SHM equation `x = A cos(2pi/T t + phi)`

we get

Amplitude, A = 1

Phase angle, `phi = - pi/6 = - 30^@`

Angular velocity, `omega = (2pi)/T = 1 "rad/s"`

The motion of the particle can be plotted as shown in the following figure.

c) `x = 3 sin (2pit + pi/4)`

`= -3cos[(2pit + pi/4) + pi/2] = - 3 cos(2pit + (3pi)/4)`

If this equation is compared with the standard SHM equation `x = Acos((2pi)/T t + phi)`, then

we get:

Amplitude, *A* = 3 cm

Phase angle, `phi = (3pi)/4 = 135^@`

Angular velocity, `omega = (2pi)/T = 2pi "rad/s"`

The motion of the particle can be plotted as shown in the following figure.

d) *x* = 2 cos π*t*

If this equation is compared with the standard SHM equation `c = Acos(2pi/T t + phi)`

#### Solution 2

a) `x = 2 cos (3t + pi/3 + pi/2)`

Radius of the reference circle, r = amplitude of SHM = 2 cm

At t = 0, `x = -2 sin pi/3 = (-2sqrt3)/2 = -sqrt2 cm`

Also `omegat = 3t :. omega = 3 "rad/s"`

`cos phi_0 = -sqrt(3)/2, phi_0 = 150^@`

The reference circle is thus as plotted below

b) `x = cos (t - pi/6 )`

Radius of circle, r = amplitude of SHM = 1 cm

At `t = 0, x = cos pi/6 = sqrt3/2cm`

`cos phi_0 = sqrt3/2, phi_0 = pi/6`

The reference circle is thus as plotted below

c) `x = 3 cos (2pit + pi/4 + pi/2)`

Here , radius of reference circle, r = 3 cm and at t = 0, `x = 3 sin pi/4 = sqrt3/2 cm`

`omegat = 2pit => omega = 2 pi "rad/s"`

`cos phi_0 = sqrt(3/2)/3 = 1/sqrt2`

Therefore the reference circle is being show below

d) `x = 2 cos pit`

Radius of reference circle, r = 2 cm and t = 0, x = 2cm

`:. omegat = pit, or omega = pi "rad/s"`

`cos phi_0 = 1, phi_0 = 0`

The reference circle is plotted below