Calculating Work Done by Variable Force on a Frictionless Semicircular Surface

[Peach]

Homework Statement

A variable force F_vec is maintained tangent to a frictionless, semicircular surface View Figure . By slowly varying the force, a block with weight w is moved through the angla theta, and the spring to which it is attached is stretched from position 1 to position 2. The spring has negligible mass and force constant k. The end of the spring moves in an arc of radius a.

Calculate the work done by the force F_vec.

Homework Equations

K_1 + U_1 + Wother = K_2 + U_2

The Attempt at a Solution

I'm not sure if I'm getting this correctly. Besides the elastic force, there's also the varying force correct? So that should Wother?

This is my equations so far:

K_1 = 0
U_1 = 0

K_2 = 1/2mv^2
U_2 = 1/2k(a*theta)^2

But I'm stuck on finding the velocity for K_2 in terms of a and theta. Any help is greatly appreciated, thanks. ... Assuming I'm correct so far (which I sort of doubt).

 

[gabee]

Actually, when the problem says it is moved very slowly, that means you can neglect the fact that it has a tiny kinetic energy. So, K_2 should be 0. The other parts of your work look correct. :)

 

[Peach]

I entered that and it says it's incorrect. What else am I missing? What about the weight on top of the spring? Does that have something to do with it?

 

[gabee]

Ah, yes, if the block is being raised vertically you will need to include potential energy of mgh after it is raised. (Of course you'll need to express mgh in terms of w and theta.)

 

[Peach]

So I have to add the potential energy of the box to the potential energy of the spring (which is the elastic force right)? So mgh_box = (w*a*theta)?

 

[gabee]

You do have to add the potential energy of the box to that of the spring, but be careful how you do it. Remember that h is the height the object has been raised, not the distance it has traveled along the semicircle.

 

[Peach]

The height isn't theta, is it? Wait, shouldn't the height be the radius then? >< Sorry I'm so confused about this circle thing.

 

[gabee]

It's okay, to see how much the block has been raised, draw a line on the diagram to represent the height the block has moved:

http://img255.imageshack.us/img255/3154/yffigure740st3.jpg

Look familiar?

 

[Peach]

Got it. So the total potential energy should be:

deltaU = wasin(theta) + 1/2k(a*theta)^2

Right? Just making sure because I only have one last attempt.

 

[gabee]

That looks good to me. :)

 

[Peach]

The answer is wrong. What else could we be missing? This is so frusterating. ><

 

[gabee]

I'm not sure. I just worked the problem by integrating an infinitesimal amount of work dW = F*dx and I got the same answer.

Net force acting on the block during its movement along the semicircle (mgsin(theta) and N cancel):
$$\sum F_{net} = F - kx - wcos(\theta) = 0$$

So,

$$F = wcos(\theta) + kx$$

(Remember x = a*theta and dx = a*dtheta)

$$dW = F dx \\ = (wcos(\theta) + kx) dx \\ = (wcos(\theta) + k(a \theta)) a d\theta \\$$

$$W = \int_0^{\theta_2} w\,cos \theta \,a\,d\theta\,+\,\int_0^{\theta_2} k a \theta\,a\,d\theta \\ = w\,a\,\int_0^{\theta_2} cos \theta \,d\theta\,+\,k a^2 \int_0^{\theta_2} \theta\,d\theta \\$$
$$W = w a sin(\theta_2) + \frac{1}{2}k a^2 \theta_2^2$$

where $\theta_2$ is the final angle (the initial angle is zero).

...which should be the correct answer...?

Is the homework an online assignment where you have to fill out a textbox? Could the error be in the format you entered?

 

[Peach]

Yeah, it's an online hw. Erm, I just checked again the answer I entered and I didn't make a mistake entering it. I don't know what's the answer since I've maxed out my attempts already. :x I'll ask my TA if it's possible. Thanks for helping me so much already, really appreciated it.

 

[gabee]

Sure, I'd be interested in knowing the answer to this problem, so keep me posted when you find out.

 

[enter260]

Peach. The answer is correct. Check the way you entered it. Make asin(theta), apparently, is not the same as a*sin(theta). The system probably screwed you over that way.

 

[Peach]

I don't remember how I entered it, whether it was asin(theta) or a*sin(theta) but the answer that I entered is showing up as asin(theta).

I can't copy it directly but this is what it's showing up as:

(w*asin(theta)) + (1/2)*k*(a*theta)^2

 

[cycam]

is there possibly a different way to write the solution? I've thought about this problem ALOT and although the answer is right, i know I'm missing something stupid.

 

[enter260]

i had the same problem as you. i entered it with the asterisk and it was fine. i believe the system interprets "asin(theta)" as arcsin?