Question

The position function x (6.0 m) cos[(3p rad/s)t p/3 rad]gives the simple harmonic motionof a body. at t 2.0 s, what are the(a) displacement, (b) velocity, (c)acceleration, and (d) phase of the motion? also, what are the (e)frequency and (f) period of the motion?

Answer 1

The correct formula for this simple harmonic motion is:

x = (6.0) cos [ (3 pi) t + pi/3 ]

A. Calculating for displacement x at t = 2:

x =6.0 cos (19/3 pi)
x = 6 (0.5)

x = 3 m

B. Velocity is equivalent to the 1st derivative of the equation. v = dx / dt

Velocity v = dx/dt = -18 pi sin (3pi t + pi/3)

v = - 18 pi sin (6pi + pi/3)

v = 49.0 m/s

C. Acceleration is the 2nd derivative or the 1st derivative of velocity dv / dt
Acceleration a = dv/dt = -54 pi^2 cos(3pi t + pi/3)

a = -54 pi^2 cos(3pi * 2 + pi/3)

a = 386.34 m/s^2

D. Phase = pi/3

E. frequency f= 3pi/2pi = 1.5Hz

F. period = 1/f = 1/1.5 = .6667sec