# Simple Harmonic Motion (will be the death of me)

1. Dec 21, 2009

### -Chad-

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

(displacement)=Asin(freq.*t+phase)

3. The attempt at a solution
"(27.1)(sin((16x)+.7))" into calculator. Where y=displacement and x=time.
I calculate the first maximum to be at (5.58,27.1). Since the question asks for the time my answer would be 5.58 seconds. But this is wrong, and I'm not sure where I went wrong..

Any help greatly appreciated, Thanks!

2. Dec 21, 2009

### rl.bhat

Hi -Chad-, welcome to PF.
At the maximum the velocity is zero. So find dx/dt and equate it to zero. From that find arc cos to find t.

3. Dec 21, 2009

### rock.freak667

How did you get t=5.58s ? If I put that value into the equation for x, I don't get 27.1. (Unless my calculator is messing up)

4. Dec 21, 2009

### -Chad-

Does 3.196 s sound like a better answer?
I changed my calculator from degrees to radians. (if this fixes my problem, sorry for the stupid mistake)

If not, rl.bhat, I do not understand what you want me to do. If I take the derivative at the maximum, where velocity is zero, it will also be zero.

5. Dec 21, 2009

### rl.bhat

In the problem dx/dt = (27.1m)(16.0)cos(16*t + 0.7) = 0.
Now at what angle cosθ = 0?
Equate that angle to 16*t + 0.7 to find t.

6. Dec 21, 2009

### ideasrule

How did you get 3.196 s? Derive x = (27.1 m) sin(16.0 t /s + 0.7) and you'll get the equation for velocity. Set the velocity to 0 and you'll be able to solve for t.

7. Dec 21, 2009

### -Chad-

?? ok, first rl.bhat, cos90=0 90=16t+.7 89.3=16t 5.58=t (same answer i already got)

ideasrule, i get the same answer when doing what you want me to do. I'll write out my derivation just for thoroughness.

________________d/dt[(27.1)(sin(16t+.7))]
product rule______(27.1)(d/dt[sin(16t+.7)])+(d/dt(27.1))(sin(16t+.7))
d/dt(sinx)=cosx___(27.1)(cos(16t+.7))+(0)(sin(16t+.7))
________________27.1cos(16t+.7)

27.1cos(16t+.7)=0
cos(16t+.7)=0
cos^-1(cos(16t+.7))=cos^-1(0)
16t+.7=90
16t=89.3
t=5.58

8. Dec 21, 2009

### denverdoc

Try using radians, 16t+0.7 = Pi/2

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