# Time of maximum displacement in simple harmonic motion?

#### liltiga

I am completely and totally lost...

1. The problem statement, all variables and given/known data
The motion of an object is simple harmonic with equation of motion x = (27.1 m) sin(16.0 t /s + 0.7). At what time after t = 0 will the displacement reach its first maximum (where velocity equals zero)?

2. Relevant equations
x = Asin(wt)

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#### pgardn

I am completely and totally lost...

1. The problem statement, all variables and given/known data
The motion of an object is simple harmonic with equation of motion x = (27.1 m) sin(16.0 t /s + 0.7). At what time after t = 0 will the displacement reach its first maximum (where velocity equals zero)?

2. Relevant equations
x = Asin(wt)
x = Asin(wt +....) The 0.7 means something in this equation... And do you know what the maximum displacement is from the equibrium position by just looking at the equation you were given?

Another way of looking at it... Do you have an equation for the velocity at any given time?

#### liltiga

oh, sorry it's x=Asin(wt+f), I think.
No, it didn't give me the displacement from equilibrium, I gave all that I had for it.

#### denverdoc

Another way of asking the same question is the equation consists of two parts:
A, the amplitude (maximal displacement) and a time varying function given by sin (some function).

Since A = 32 = constant, the maximal value will occur when sin (some function) = 1. This of course will happen over and over as sin varies between -1 and 1. But it is asking for the value of t when this first occurs. So what you need to solve for is when the argument of the sin is equal to pi/2 radians(90 degrees). In other words (16t + 0.7) = pi/2.

The other approach using knowledge of min, max and first derivatives is to take the derivative of the function 27.1 sin (16t+0.7). and solve for when that will be zero.

#### pgardn

oh, sorry it's x=Asin(wt+f), I think.
No, it didn't give me the displacement from equilibrium, I gave all that I had for it.
Well actually its A, as long as the entire part sin(wt + f) = 1.

But because your mass or whatever is at a starting position of f, then the mass is going to move in some amount of time to get to A. f is some starting position, the position at t = 0s.

So I have given you one hint with the first equation on how to find the time where sin() = 1 meaning the object would be at its maximum displacement from equilibrium.

Another way is to get the velocity equation with respect to time. You have the position equation with respect to time. So if you know the velocity equation, you can set V(t) = 0 m/s or w(t) = 0 rad/s and get the time where the object that is oscillating is at its maximum displacement. Have you been given the velocity equation for SHM, or have you been shown how to take the derivative of the position equation with respect to time to get the velocity function with respect to time for SHm?

Whoops I see the other poster has said somewhat the same thing.

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