Question

If the equation of motion of a particle is given by s = A cos(ωt + δ),
the particle is said to undergo simple harmonic motion. s'(t) = -Aω sin(ωt + δ)

When is the velocity 0? (Use n as the arbitrary integer.)

t= _____________

I thought it was 0, but it said it was wrong.

Answer 1

The solution to t in the following equation is the right answer.

s'(t) = -Aω sin(ωt + δ) = 0

Since it is an SHM (simple harmonic motion), there are many solutions.

Think of riding on a swing. When you reach the highest point, the velocity is zero. But <i>every</i> time you ride to the top (on each side of the swing), the velocity is zero. So this happens every 2 seconds or so. The arbitrary integer n is used to denote the sequence.

Back to the equation:

s'(t) = -Aω sin(ωt + δ) = 0

Since we know the amplitude A is not zero, and the frequency ω is not zero, so sin(ωt + δ)=0
The solution of
sin(ωt + δ)=0
is
ωt + δ)= nπ
(see link at end of post)
So solve for t to get:
t=(nπ-δ)/ω


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