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.
(a) Find the velocity of the particle at time t.
s'(t) = - A ω sin(ωt + δ)
I figured out how to do part a, but I don't know how to solve for part b.
(b) When is the velocity 0? (Use n as the arbitrary integer.)
t = __________
To solve part b, we need to find the values of t for which the velocity of the particle is 0.
We are given the equation for velocity:
s'(t) = - A ω sin(ωt + δ)
For the velocity to be 0, sin(ωt + δ) must be equal to 0. In other words, we need to solve the equation:
sin(ωt + δ) = 0
To find the values of t that satisfy this equation, we can use the fact that sin(θ) = 0 when θ is an integer multiple of π:
ωt + δ = nπ
Here, n represents an arbitrary integer. Now, let's solve for t. First, isolate t:
ωt = -δ + nπ
Then, divide both sides by ω:
t = (-δ + nπ) / ω
Therefore, the values of t for which the velocity is 0 are given by:
t = (-δ + nπ) / ω, where n is an arbitrary integer.
Note that n can take on any integer value, both positive and negative, to capture all the possible solutions.