A body oscillates with simple harmonic motion according to the equation
X = 6.0 cos(3t + $\pi$/3)
Where the units are SI. The speed of the body when it has a displacement of
3 m is
A) m/s
B) 6$\pi$m/s
C) 9$\pi$ m/s
D) m/s
E) 18$\pi$ m/s

Question

A body oscillates with simple harmonic motion according to the equation
X = 6.0 cos(3t + $\pi$/3)
Where the units are SI. The speed of the body when it has a displacement of
3 m is
A) m/s 
B) 6$\pi$m/s 
C) 9$\pi$ m/s 
D) m/s 
E) 18$\pi$ m/s

Quizplus · Accepted Answer

We can find the velocity of the body by taking the derivative of the displacement equation:

v = dx/dt = -18 sin(3t + π/3)

To find the velocity when the body has a displacement of 3 m, we need to plug in X = 3 to this equation and solve for v:

3 = 6 cos(3t + π/3)
cos(3t + π/3) = 1/2
3t + π/3 = ±π/3 + 2nπ   (where n is an integer)
t = (±π/3 - π/3 + 2nπ)/3
t = (±π/9 + nπ/3)

Plugging these values of t into the velocity equation, we find:

v = -18 sin(±π/3) = ±9√3 m/s

There are two possible values for v because there are two values of t that correspond to X = 3 (when the body is at maximum displacement, and when it is returning to that position).

However, we are only asked for the speed (magnitude of velocity), which is the same for both values:

|v| = 9√3 m/s

This matches choice D.

A Body Oscillates with Simple Harmonic Motion According to the Equation