1. ## Simple Harmonic Motion

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

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

(a) When $\displaystyle t = 0, y = 2\sin(0 - \pi/2) = -2$. 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 $\displaystyle t = 3, y = 2\sin(3\pi - \pi/2) = 2\sin(5\pi/2) = 2$. So the mass is $\displaystyle 24-2=22$ feet below the ceiling at $\displaystyle t=3$.

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

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

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

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

3. 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.

Thanks

4. 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.

Yes, as I said at the beginning, I am assuming that $\displaystyle y$ is measured positive upwards. If it's positive downwards, then the value of $\displaystyle y$ 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 $\displaystyle 24 + (-2) = 22$ feet, and in part (b) the distance when $\displaystyle t = 3$ is $\displaystyle 24 + 2 = 26$ feet.
Read the question you were given again, and study the wording carefully. It should make it clear which way $\displaystyle y$ is to be measured.