Finding the Work Done in Moving a Mass on a Half-Cylinder at Constant Speed

[Ailo]

Homework Statement

A small mass m is pulled to the top of a frictionless halfcylinder (of radius R) by a cord that passes over the top of the cylinder. (a) If the mass moves at a constant speed, show that $$F=mg cos(\theta)$$. The angle is between the horizontal and the radius drawn to the mass.

(b) By directly integrating
$$\int{Fds}$$
find the work done in moving the mass at constant speed from the bottom to the top of the half-cylinder. Here ds represents an incremental displacement of the small mass.

Homework Equations

The Attempt at a Solution

The a-part was easy when I drew a diagram. The b-part is the one I'm struggling with. With the Work-Energy theorem I get that the work done by F is mgR. But what integral should I compute and why? Have no idea whatsoever..

 

[tiny-tim]

(b) By directly integrating
$$\int{Fds}$$
find the work done in moving the mass at constant speed from the bottom to the top of the half-cylinder. Here ds represents an incremental displacement of the small mass.
…
The a-part was easy when I drew a diagram. The b-part is the one I'm struggling with. With the Work-Energy theorem I get that the work done by F is mgR. But what integral should I compute and why? Have no idea whatsoever..

Hi Ailo!

(I'm not sure what you mean by mgR)

The integral is given to you … ∫F ds, where s is the displacement.

(Remember, the string is always tangent to the cylinder.)

 

[Ailo]

Hi! Thx for the answer, but the problem is how to set it up. I should get an expression, integrate it, and end up with the answer mg*R. I've got the force as a function of the angle, and I don't understand how to integrate it over a distance.

Maybe I didn't explain the situation good enough. The half cylinder lies on the ground, and we pull the mass up along the quartercircle. Does anybody understand? =)

 

[tiny-tim]

Hi Ailo!

Just decide what ds is (in terms of θ), and then integrate mgcosθ ds, and you should get mgR.

 

[Ailo]

That's the problem. I've never done a problem like this before...

 

[tiny-tim]

ok …

what is ds in terms of θ?

in other words, if you increase the angle by dθ, how much do you increase the length (s) of the string by?

 

[Ailo]

My best guess is to make a triangle with sides R, R and ds. Will that work?

 

[Ailo]

Ohh! Now I get it. It's (theta)*R, right?

*palmslap

 

[tiny-tim]

Yup!

ds = Rdθ