Constant velocity on frictional slope

[Loren Booda]

Derive the slope of a track where a sphere having frictional coefficient u maintains constant velocity v.

 

[Tide]

Show us what you've tried so far.

 

[Loren Booda]

Tide,

It's not a homework problem (yet?), but I appreciate your response. I thought someone might have fun working out this nontrivial, simply stated puzzle.

 

[Gamma]

If I use Torque = I * alpha and Newtons second law, I end up with the following for acceleration.

a = mg sin(theta) * r^2 /(I + mr^2) where I is w.r.t. the center of the rolling object.


If I use the energy relation (i.e Change in Energy = work done by the force), I get

a = g[2 sin (theta) + u cos(theta)] / ( 1 + I/mr^2)

I can't figure out where the discripancy is.

 

[Doc Al]

If I use Torque = I * alpha and Newtons second law, I end up with the following for acceleration.

a = mg sin(theta) * r^2 /(I + mr^2) where I is w.r.t. the center of the rolling object.

Assuming the sphere rolls without slipping, this looks good.

If I use the energy relation (i.e Change in Energy = work done by the force), I get

a = g[2 sin (theta) + u cos(theta)] / ( 1 + I/mr^2)

Not sure what you did here. Realize that as long as the sphere rolls without slipping, the friction does no work.

 

[Gamma]

Say that the sphere rolls down from top a distance x.

Final Energy Ef= 1/2 I w^2 + 1/2 m v^2 - mgx sin(theta). Of course if the energy is conserved I can show that accelaration 'a' is same as the first expression in my post which is correct for a rolling object down the hill.

So my question is what effect do the gravitational and frictional forces have on the rolling motion? What makes the ball roll?

 

[Doc Al]

If there were no friction, the ball would slide down the hill without rolling. Friction exerts the torque (about the ball's center of mass) that makes the ball rotate as well as translate. As long as the friction is sufficient to make the ball roll without slipping, the friction does no work and mechanical energy is conserved. Friction enables the gravitational PE to be transformed into both rotational and translational KE.