Springs and potential/kinetic energy

[Carrie]

Homework Statement

A 0.50-kg mass is attached to a spring of spring constant 20 N/m along a horizontal, frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position.

(a) What is the amplitude of vibration?
I already solved for the amplitude and got 0.24 meters.

(b) At what location are the kinetic energy and the potential energy the same?
?

Homework Equations

E = 1/2mv^2 + 1/2kx^2

The Attempt at a Solution

E = 1/2mv^2 + 1/2kx^2

I thought I would set the kinetic energy and potential energy equal to each other, so I got:

1/2(.5)(1.5)^2 = 1/2(20)x^2
From that, I got x = .24 meters.

But that's wrong and the answer is 0.17 meters. I'm confused as to where I went wrong. :(

Thank you!

 

[TSny]

I thought I would set the kinetic energy and potential energy equal to each other, so I got:

1/2(.5)(1.5)^2 = 1/2(20)x^2

Why did you use 1.5 m/s for the speed here?

 

[Carrie]

Oh, the 1.5 m/s is for the equilibrium position, which I guess isn't necessarily the same position?

 

[TSny]

Right. The location where the KE equals the PE is not the equilibrium position.

 

[Carrie]

I'm sorry, but now I'm really lost. How do I find v then, or am I using the wrong the wrong equation or something?

 

[TSny]

What can you say about the total energy E at different points of the motion?

 

[Carrie]

The total energy stays the same.

 

[TSny]

The total energy stays the same.

Yes. If you pick any two points of the motion, then E is the same for those two points. You probably used that idea when answering part (a).

See if you can use the same idea to help solve (b).

 

[Carrie]

Ohhh okay, so if at equilibrium, the total energy (when PE is 0) is 1/2 mv^2, which is 0.56 J, then when they're the same:

0.56 (total energy in the system) = 2* 1/2kx^2 - I'm not sure if the theory is right, but since KE = PE, then KE+PE becomes PE + PE and so you can multiply by two. Maybe. Because I did get the answer this way, but I also want to make sure I didn't just get lucky.
x = 0.17 m.

 

[TSny]

You are thinking correctly. Good work.

 

[Carrie]

Awesome! Thank you so much for your help!