Friction and Torque
Date: Fall 2013
If a sphere is placed on an incline that is non-friction free a torque is created by the friction that causes the sphere to rotate. The torque applied is the product of the force of static friction and its magnitude is equal to the force of friction x the torque arm; the torque creates a constant angular acceleration. I am confident in these statements however....If the coefficient of static friction, and thus the force of static friction, is increased it would reason that the torque is increased proportionally as well as the angular acceleration. Conceptually that would imply that a greater static friction force would create a greater angular acceleration and the sphere would actually roll faster down the incline. Empirical evidence does not support this finding. Is there a fault in my reasoning? What effect DOES increasing the coefficient of friction have on torque, angular acceleration, and time down the incline.
While I cannot get into the rotary moment of inertia of the sphere, I can tell you that the friction presents only an upper limit to the torque. So long as friction is sufficient so that the sphere is only rolling and not sliding on the incline, the friction should not affect the result. However if the friction is small enough to allow slippage, then the rotational acceleration will no longer be proportional to the linear acceleration. The result would then be increased linear acceleration and reduced rotational acceleration.
If you can imagine driving a car on a somewhat slippery surface (such as a smooth steel plate) the reduced friction will not affect the motion of the car so long as the applied force (from ste
ering, acceleration, and braking) does not overcome the reduced friction. This example is similar.
First: You have to examine your assumption. You say: ?on an incline that is non-friction free a torque is created by the friction that causes the sphere?. If there is no friction between the sphere and the incline then there is no interaction. Then the trajectory of the sphere has no angular momentum, and the trajectory is determined by gravity. In this case, it does not matter whether the sphere is stationary or rotating with respect to an external frame of reference.
You cannot delete friction as a force, then turn it off, then turn it on again. You cannot turn on and turn off friction. ?The on an incline that is non-friction free a torque is created by the friction that causes the sphere.
The torque is created by the linear acceleration of the sphere as gravity pulls it down the hill. If the coefficient of friction were zero, the sphere would slide without rotating, and all of its potential energy would be converted into translational kinetic energy. If the coefficient of friction were so large that the sphere could not slide even a little bit, its potential energy would be converted partly into translational and partly into rotational kinetic energy, so the sphere would roll more slowly downhill, and the translation and rotation would be strictly coupled. If the sphere could slide a little, the rotation speed would increase slowly until it matched the translation speed.
How do you figure out what fractions of the potential energy go into translational and rotational kinetic energy? The easy way is to write down the Lagrangian, and solve the Euler-Lagrange equation. This is one of the standard example problems for Lagrangians in a college-level classical mechanics course, so you can probably find the solution in detail somewhere on the web. Wikipedia has some nice stuff on this, for example.
Here's your error: ".If the coefficient of static friction, and thus
the force of static friction, is increased..."
The coefficient of static friction describes the *maximum* force that
an object can resist before sliding, but the force applied by the
surface is simply "enough" to prevent motion -- in other words, it is
equal (and opposite) to the force applied by the object, and *not* a
function of the coefficient of static friction. Increasing the
coefficient of static friction does not increase the force applied by
the surface in examples where the ball would roll. The only time the
coefficient of static friction gets involved is when the force exceeds
the threshold of force possible with static friction (which *is* a
function of the coefficient of static friction) and turns into a
sliding/rolling combination (kinetic) friction problem. In those
examples, increasing the force of static friction might cause the ball
to return to a rolling-only example, instead of sliding/rolling.
If you were to change the angle of the incline, or change the force of
gravity, or reduce the coefficient of static friction, you could
achieve examples where the ball slides, rather than rolls. In these
examples, the coefficient of kinetic friction would govern the force,
not the static friction - there would be a momentary force governed by
the static friction, that would immediately drop, according to the
kinetic friction coefficient. The only example where the force
experienced by the ball is a function of the coefficient of static
friction is where the force experienced is precisely at the cusp of
sliding -- that is the threshold of static friction.
Hope this helps,
The torque is only going to increase through the addition of friction as long as the ball is slipping. Once the ball is rolling without slippage I do not think you will get any further increase in torque. Where would it come from if you did? Since friction is the force that tries to keep the ball from skidding along, once the ball rolls freely the static friction is not doing much.
To put it another way, the fastest friction can get the ball rolling will be at the point where the ball is no longer impeded by friction. Now maybe if you had the ball rolling on a very sticky surface you could stretch out the range where friction continues to affect the rotation of the ball. On a hard surface static friction only effects the object when it first begins to move. After that there are some dynamic friction forces that tend to oppose movement--like wind resistance.
The other limiting factor is acceleration due to gravity, which is the force propelling the ball in the experiment. The movement of the ball is impeded as long as static friction prevents the ball from rolling freely under the force of gravity. When static friction no longer has an effect you will get the maximum effect of gravity on the rotation of the ball.
I hope I have correctly understood your question, and that this helps a little.
A greater static friction coefficient does not result in a greater static friction force. It only affects the maximum possible static fiction force. So long as the maximum possible value is not surpassed, the static friction force is only as large as is necessary to prevent the surfaces from sliding. Although the kinetic friction coefficient sets the force, the static friction coefficient sets the limit.
Dr. Ken Mellendorf
Illinois Central College
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