Work Done on Block Moving in Circle with Changing Radius

[Devin Longo]

Homework Statement

A small block of mass 0.91 kg slides without friction on a horizontal table. Initially it moves in a circle of radius r0 = 0.63 m with a speed 1.5 m/s. It is held in its path by a string that passes through a small hole at the center of the circle. The string is then pulled down a distance of r0 - r1 = 0.12 m, leaving it at a radius of r1 = 0.51 m. It is pulled so slowly that the object continues to move in a circle of continually decreasing radius.
How much work was done by the force to change the radius from 0.63 m to 0.51 m?

Homework Equations

W net = mv f 2 - mv o 2

The Attempt at a Solution

I tried to use the work energy theorem, but I'm confused because I think I'm getting the wrong numbers? Can anybody guide me?

 

[Delphi51]

This looks like an interesting one! It should be possible to work it out using either energy or forces. But first you must use conservation of angular momentum to find how fast the block is going at the final radius.

 

[rl.bhat]

m*v^2/r is the force required to keep the block in the circular motion.
Since you are applying the force radially by pulling the string, the speed will remain constant. String pulls the block towards the center. As a reaction block pulls the string away from the center. If you move the block through a distance dr towards the center by pulling the string, the work done dW = - (m*v^2/r)*dr.
To find the net work, find the integration from r = r0 to r = r1.

 

[Delphi51]

Your expertise is legendary, rl! Can you help me understand why speed is constant rather than angular momentum in this situation? I guess I'm thinking of the skater spinning and pulling her arms in - she speeds up.

 

[rl.bhat]

There is no friction on the table. The speed will change only when a force acts tangentially. Such force is absent in this problem. In the skater case there is no external force acting on her.

 

[Delphi51]

There is no friction on the table. The speed will change only when a force acts tangentially. Such force is absent in this problem. In the skater case there is no external force acting on her.

That seems to make sense . . . but why would the angular momentum change when there is no torque? Just for fun (can't let the students have all the fun), I worked it out both ways and the constant velocity method doesn't seem to make sense to me energy wise:

I'm ignoring the minus signs on the work calcs - the force should have been reversed to get the point of view of the string pulling the block.

 

[rl.bhat]

You are absolutely right. Thank you.

 

[Delphi51]

Okay! Now if we could only figure out why the speed increases when there is no tangential force . . . perhaps something to do with the non-circular path of the block? As it is spiraling (?) inward, there is a component of velocity in the radial direction which the force increases. Perhaps both approaches are correct and in my constant (tangential) velocity solution I failed to include the radial velocity and its energy.

Fascinating problem - hats off to the author.