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- Thread starter fobos3
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Doc Al

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Yes. And the required static friction for rolling without slipping depends on the angle. And the maximum available static friction also depends on the angle and the coefficient of friction.If we have an inclined plane at angle [tex]\alpha[/tex] and a sphere on that plane it will have angular acceleration due to the friction force.

That's not the exact relationship, but your overall idea is correct. If the maximum available static friction is insufficient to create the needed torque, the sphere will slide as well as roll. (I think that's what you are getting at.)If [tex]F_{lim}<mg\sin \alpha[/tex] , where [tex]F_{lim}[/tex] is the limiting friction will the sphere also slide along the plane?

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rcgldr

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To simplify things you could assume that static and dynamic friction are the same and independent of speed. Assuming that friction force is less than mg sin(theta), and an experiment that operates in a vacuum, then the rate of both linear and angular acceleration would be linear. If the friction force equals mg sin(theta), with an initial linear velocity, the linear velocity would remain constant, but there would a be a constant rate of angular acceleration (until the sphere started rolling in which case both linear and angular acceleration would occur due to the reduced opposing friction force once the sphere starts rolling).

Depending on angular inertia, initial velocity, and coefficient of dynamic friction, if the rate of angular acceleration times radius is greater than the rate of linear acceleration, eventually the sphere starts rolling without sliding. Take the simple case of a sphere sliding from a frictionless horizontal plane to a non-frictionless horizontal plane. Then angle the plane until it reaches the point where tan(theta) = coefficient of dynamic friction where the ball slides at constant speed but increases angular velocity until it starts rolling. Once tan(theta) > coefficient of dynamic friction, then both linear and angular velocity increase and if angular inertia and/or angle of plane is high enough (all the mass at the surface of the sphere, like a ping pong ball), the sphere may never transition into rolling.

Depending on angular inertia, initial velocity, and coefficient of dynamic friction, if the rate of angular acceleration times radius is greater than the rate of linear acceleration, eventually the sphere starts rolling without sliding. Take the simple case of a sphere sliding from a frictionless horizontal plane to a non-frictionless horizontal plane. Then angle the plane until it reaches the point where tan(theta) = coefficient of dynamic friction where the ball slides at constant speed but increases angular velocity until it starts rolling. Once tan(theta) > coefficient of dynamic friction, then both linear and angular velocity increase and if angular inertia and/or angle of plane is high enough (all the mass at the surface of the sphere, like a ping pong ball), the sphere may never transition into rolling.

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If the slope angle is theta, the force

F

The force

F

If the coefficient of friction is C

F

If the mass were a point then the total kinetic energy is

-mgh = 1/2 m v

I = (2/5) m R

- mgh = (7/10) m v

So there is an opposing frictional force slowing down the acceleration. By equating the two equations, the actual (effective) downhill acceleration force is F = 5/7 mg sin (theta). Thus there must be an uphill frictional (opposite to mg sin(theta)) force

F

So the ball will slide if (2/7) mg sin(theta) > C

or if

tan(theta) > (7/2) C

This is a WAG (wild *** guess).

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Ya. i got the same result. 7/2 Cf. So it's a SAG rather.

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