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 (3t + π/3)

(b) x = cos (π/6 – t)

(c) x = 3 sin (2πt + π/4)

(d) x = 2 cos πt

Answer 1

(a) x = -2sin(3t + π/3)
= 2cos(3t + π/3 + π/2)
= 2cos(3t + 5π/6)
Compare this equation to standard equation , x = Acos(wt + ∅)
A = 2 cm
w = 3 rad/s
∅ = 5π/6
See the figure here we graph of it .


(b) x = cos(π/6 - t)
= cos( t - π/6)
compare this equation then,
∅ = -π/6
A = 1 cm
w = 1 rad/s
See figure above .

(c) x = 3sin(2πt + π/4)
= - 3cos( 2πt + π/4 + π/2) [cos(π/2+∅) = -sin∅]
= -3cos(2πt + 3π/4)
= 3cos( 2πt + 3π/4 +π)
= 3cos(2πt + 7π/4)
Compare the standard equation ,
A = -3cm
w = 2π rad/s
∅ = 7π/4 rad
See the figure .


(d) x = 2cosπt
A = 2
w = π
Phase angle = 0
See the figure .