Max Ramp Angle for Rolling Ball: mu & Theta

[rhuala]

Ball mass m and radius r rolls down ramp with coefficient of static friction of mu. If the ball is released from rest what is the maximum angle theta of the ramp that the ball rolls without slipping?

I've got theta max = inverse(tan(mu)) but the answer in the book is

theta = inverse(tan(3.5mu))

I'm not sure where the 3.5 comes in could someone please explain?

Also if 2 balls roll down a ramp, one is filled with fluid the other not which one reaches the bottom first. The mass cancels out the the equation for the acceleration so it seems to me they should reach the bottom at the same time. Just wanted to verify that this is correct.

Thanks in advance

Carla

 

[DaveC426913]

"Also if 2 balls roll down a ramp, one is filled with fluid the other not which one reaches the bottom first. "

Does the fluid-filled one weigh more?

It seems to me that the lighter ball would reach the ground first.
1] The lighter ball will alternate between rolling and freefalling (after each little bump)
2] The The heavier ball also resists the turning caused by friction, thus will accelerate slower.

 

[Doc Al]

I've got theta max = inverse(tan(mu)) but the answer in the book is

theta = inverse(tan(3.5mu))

I'm not sure where the 3.5 comes in could someone please explain?

Perhaps you are confusing this problem with finding the maximum angle that an incline can be increased before an object begins sliding down?

Hint: There is a net force acting down the incline---apply Newton's 2nd law for both translation and rotation.

 

[rcgldr]

Friction and viscosity of the fluid should cause the fluid filled ball to roll slightly slower.

 

[rhuala]

Friction and viscosity of the fluid should cause the fluid filled ball to roll slightly slower.

Can you (or someone) show this through formula please, I'm not sure this is correct...

 

[rcgldr]

I don't know the formula's for this, but here's an similar example.

Replace the fluid with an object of a certain mass with a low coefficient of friction. As the ball rolls, the object is raised a bit, and then starts sliding inside of the ball as the ball rolls. In addition to increasing the kinetic engergy of the ball, temperature energy is also being increased (from the friction).

Since the initial potential engery is the same for both balls at the start, the ball with the increasing temperature energy ends up with less of an increase in kinetic energy.

 

[vincentchan]

ignore what jeff said... air friction play no role in this problem
identify the problem and read #3 post carefully... let me know how far you get or where you stuck b4 i can further help you... this problem is not as hard as you think... and don't expect it is a easy problem... (am i contradicting myself?)

 

[rcgldr]

ignore what jeff said... air friction play no role in this problem

I never mentioned air friction. One of the questions concerened fluid in a ball, which is a source of internal friction (viscosity).