Help a Sinking Physics Student: Maximum Energy & Velocity Problems

[ph0bolus]

Homework Statement

1)
A 1.10 kg mass on a spring oscilates horizontally with little friction according to the following equation: x=0.050cos(2.9t), where x is in meters and t in seconds. Find the maximum energy stored in the spring during an oscillation.

2)
Find the maximum velocity of the mass.

3)
A 55.0 kg circus performer oscillates up and down at the end of a long elastic rope at a rate of once every 9.80 s. The elastic rope obeys Hooke's Law. By how much is the rope extended beyond its unloaded length when the performer hangs at rest?

Homework Equations

I think for all problems i use an equation dealing with X=Acos(wt) where w is angular freq.

The Attempt at a Solution

I've attempted the problem trying to figure out what goes where, but nothing adds up. I know the mass and time for number 1 since it's given, but what to do from there?

For problem #2 to get the maximum velocity i should use v=square root(k/m)A, but i don't know what the amplitude is.

#3 i don't even know where to get started.

If anyone could help a sinking physics student out it would be greatly appreciated.

 

[Kurdt]

For number 1 you need to know what the energy stored in a spring is. Simply the maximum energy stored will be when it is stretched or compressed to its fullest. In other words when cos(2.9t)=+/-1.

I'm sure you're aware that velocity is the time derivative of position. And in a similar way to question 1 when will velocity be maximum?

For question 3 do you know what Hooke's law is? If you do you basically have to determine the spring constant of the elastic cord. Do you know an equation that relates the time period the mass and the spring constant?

 

[ph0bolus]

For number 1 i think you're going to have to explain it a little bit more because i don't know how to work it out. So since the maximum energy stored is when it's stretched or compressed the time would have to be equal to 0 is that what you're saying?

Number 2 would the maximum velocity be when the spring is in movement? Probably not right, but if it is which equation would i use to solve for it?

3 Hookes law is f=-kx, but how do i get the spring constant if x isn't given?

 

[Kurdt]

For number 1. Energy stored in a spring is $\frac{1}{2} k x^2$. x is going to be maximum when cos(2.9t) = 1 which will give you the maximum stored energy per oscillation. You will need to find the spring constant from the angular frequency. What equations have you been given for that?

The maximum velocity occurs when the mass is at the equilibrium point. The question will be similar to question 1. I'm assuming you understand:

$$v = \frac{dx}{dt}$$

For part 3 you are given the time period of the oscillation. are you familiar with:

$$T = 2\pi \sqrt{\frac{m}{k}}$$ ?

 

[ph0bolus]

angular frequency= square root(k/m) would that be the equation I'm using?

for #2 would the equation look something like this? -Aw*sin(wt)?

i found that the spring constant= 133.79 n/m so then i plugged it into -kx=mg to try and find x and got 4.03..that's wrong right?

 

[Kurdt]

angular frequency= square root(k/m) would that be the equation I'm using?

Yes that is what you would use.

for #2 would the equation look something like this? -Aw*sin(wt)?

Yes you are correct.

i found that the spring constant= 133.79 n/m so then i plugged it into -kx=mg to try and find x and got 4.03..that's wrong right?

Yes that's different to what I get.