If I have a spool of thread with a force pulling to the right on a rolling cylinder, how can I mathematically determine the direction of the friction?
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With friction, the bottom doesn't slide. It matches the speed of the table, which is $0$. The spool rolls.
Suppose the spool was sitting on ice, so F was the only force. How would the spool accelerate? The bottom would slide. The spool would still rotate, but not as fast as with friction.
So friction holds the bottom of the spool back. It is a backward force.
This explains more about the forces involved. Toppling of a cylinder on a block
We use the concepts of Impulse and Angular impulse. If the force is applied below the center of gravity in forward direction then, frictional force will act in backward direction because the contact point has both angular velocity and translational velocity in forward direction and friction tries to make it stationary relative to the surface.
Let us apply a force F at a height "h" above the center of gravity of the cylinder. Impulse(J): mv = Ft Angular impulse = Iw = τt = Fht (h is distance perpendicular to force)= J*h Iw = mvh In the case of pure rolling, v= wr I/r = mh h = I /mr If force is applied at a distance H(> I/mr) above the center of gravity, frictional force would act in the direction of force and if H lies below I/mr, frictional force will act opposite to that of force and ofcourse parallel to surface and passing through the contact point.
And since we have a spool of thread, force is being applied at a distance r from COM, so frictional force will act in the direction of force. (ASSUMING, I < mr²) HERE, I = MOMENT OF INERTIA ABOUT COM w = angular velocity t = time m = mass τ = Torque r = radius v = velocity
I am assuming the following: The inertia of the object is that of a cylinder. Spool of thread is massless
