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.

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 every 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

http://www.google.ca/imgres?imgurl=http://intmstat.com/trigonometric-graphs/sinx.gif&imgrefurl=http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php&h=207&w=328&sz=3&tbnid=oJ2eIa-eurAqkM:&tbnh=84&tbnw=133&prev=/search%3Fq%3Dsine%2Bcurve%26tbm%3Disch%26tbo%3Du&zoom=1&q=sine+curve&docid=UDPUCTbk1CVn-M&hl=en&sa=X&ei=aY3VTofDJsXn0QGnv9yGAg&sqi=2&ved=0CDQQ9QEwAQ&dur=3321

Well, well, well. Looks like someone is experiencing a bit of a velocity conundrum. Don't worry, I'm here to help you through it with my delightful humor.

Let's take a closer look at the equation for velocity, s'(t) = -Aω sin(ωt + δ). To find when the velocity is zero, we need the sine function to be zero. And when does that happen? Ah, yes, when we have a little gathering called the "Zero Party"!

In the world of sine, the function is zero at a multitude of times. We can express these moments using an arbitrary integer, n. So, my dear friend, the velocity of your particle is zero at times t = nπ/ω - δ/ω.

Now go ahead and throw a celebratory dance party in honor of the zeros of velocity. Just make sure to mind your manners and don't let your dance moves become too harmonic!

To find when the velocity is 0, we need to equate the expression for velocity to 0:

s'(t) = -Aω sin(ωt + δ) = 0

The sine function is equal to 0 when its argument is equal to nπ, where n is an integer. Therefore, we can set the argument of the sine function equal to nπ:

ωt + δ = nπ

To isolate t, we can subtract δ and divide by ω:

t = (nπ - δ)/ω

So, the value of t at which the velocity is 0 is:

t = (nπ - δ)/ω

To find the time when the velocity is zero, we need to solve the equation s'(t) = -Aω sin(ωt + δ) = 0.

In this case, the equation s'(t) = -Aω sin(ωt + δ) represents the velocity of the particle in simple harmonic motion.

Now, let's set the equation equal to zero and solve for t:

-Aω sin(ωt + δ) = 0

Since sin(ωt + δ) can only be zero for certain values of (ωt + δ), we need to find when sin(ωt + δ) = 0.

For sin(ωt + δ) to be zero, the argument (ωt + δ) of the sine function must equal nπ, where n is an integer.

So, we have:

ωt + δ = nπ

Solving for t, we get:

t = (nπ - δ) / ω

Therefore, the time when the velocity is zero is:

t = (nπ - δ) / ω

Make sure to substitute the values of A, ω, and δ from the given equation of motion to get the final answer.