Simple Harmonic Motion

**DBA** · May 15th 2009, 10:15 PM

Hello, I need help with the following:

The motion of a spring-mass system is described by the equation y=2sin(pi*t - pi/2) where y is the distance in feet from equilibrium position and t is time in seconds.

a. If the weight is 24 feet from the ceiling in a state of equilibrium, find the starting position of the weight (time=0).

b. Find the position of the mass at 3 seconds.

c. What is the first time the mass will reach equilibrium position?

Thanks

**Grandad** · May 16th 2009, 12:56 AM

Hello DBA

Welcome to Math Help Forum!

Originally Posted by DBA

Hello, I need help with the following:

The motion of a spring-mass system is described by the equation y=2sin(pi*t - pi/2) where y is the distance in feet from equilibrium position and t is time in seconds.

a. If the weight is 24 feet from the ceiling in a state of equilibrium, find the starting position of the weight (time=0).

b. Find the position of the mass at 3 seconds.

c. What is the first time the mass will reach equilibrium position?

Thanks

I assume that $y$ is measured positive upwards, so $y$ represents the height above the equilibrium position, which is 24 feet below the ceiling. (Your statement of the question says that $y$ is the distance from the equilibrium position, which is not the same thing!)

(a) When $t = 0, y = 2\sin(0 - \pi/2) = -2$ . On the assumption I've made above, this means that the mass is 2 feet below the equilibrium position. So the starting position of the mass is 26 feet below the ceiling.

(b) When $t = 3, y = 2\sin(3\pi - \pi/2) = 2\sin(5\pi/2) = 2$ . So the mass is $24-2=22$ feet below the ceiling at $t=3$ .

(c) When $y = 0, \sin(\pi t - \pi/2) = 0$

$\Rightarrow \pi t - \pi/2 = 0, \pi, 2\pi, ...$

$\Rightarrow t = \tfrac12, \tfrac32, \tfrac52, ...$

$\Rightarrow$ the first time the mass is in the equilibrium position is after $\tfrac12$ sec.

Grandad

**DBA** · May 16th 2009, 05:05 AM

Hello, thank you very much.
The solution for part a must be 22 feet. Can you explain what the different between the two statements is (the one you assumed and the one given in the text)?

It seems that I have to substract the -2 from the 24feet and I do not understand why.

Can you please help with that.
Thanks

**Grandad** · May 16th 2009, 05:14 AM

Hello DBA

Originally Posted by DBA

Hello, thank you very much.
The solution for part a must be 22 feet. Can you explain what the different between the two statements is (the one you assumed and the one given in the text)?

It seems that I have to substract the -2 from the 24feet and I do not understand why.

Can you please help with that.
Thanks

Yes, as I said at the beginning, I am assuming that $y$ is measured positive upwards. If it's positive downwards, then the value of $y$ must be added to the distance of 24 feet to get the total distance of the mass from the ceiling.

So in part (a) the distance initially is $24 + (-2) = 22$ feet, and in part (b) the distance when $t = 3$ is $24 + 2 = 26$ feet.

Read the question you were given again, and study the wording carefully. It should make it clear which way $y$ is to be measured.

Grandad

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