When a ball rolls down a ruff slope the frictional force acts tangent to the ball and causes the angular acceleration of the ball but at the same time the frictional force is acting to reduce the translational acceleration of the ball. How is this possible when the frictional force is acting only tangentially and not through the centre of mass of the ball?
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2 Answers
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Forces don't need to act through the center of mass to produce translational acceleration. Consider a rod lying on a horizontal frictionless surface - applying a force at one end of the rod perpendicular to its axis will cause the rod to both spin and move translationally. Or, consider a rolling object placed on a rug - pulling the rug causes the object to roll and move toward you, the object doesn't just spin in place.
Forces acting through the center of mass don't induce rotation, but that doesn't imply that forces not acting through the center of mass don't induce translation.
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$\begingroup$ Take your rod (with mass m) on a frictionless surface example. If you apply a force at the end of the rod perpendicular to the rod the acceleration of the COM of the rod isn’t F/m. However with friction that is the case. All of the frictional force acting perpendicular is also applied through the COM. This difference is strange to me $\endgroup$– Blue5000Commented Aug 27, 2022 at 12:58
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$\begingroup$ @Blue5000 The rod spins whether there is friction or not. Friction is not only applied through the center of mass - if that were the case, friction could not slow the rod's rotation, since forces acting through the COM don't induce torque. But clearly, friction can stop a spinning object. $\endgroup$ Commented Aug 29, 2022 at 21:01
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How is this possible when the frictional force is acting only tangentially and not through the centre of mass of the ball?
It is possible because the static friction force that enables rolling (without slipping) gives the ball both rotational kinetic energy and translational kinetic energy of its center of mass (COM). The sum of the rotational kinetic energy and translational kinetic energy, given a ball that begins rolling from rest, equals the loss of gravitational potential energy.
If the slope were frictionless, the ball would slide down the surface without rolling and all of its KE would be the translational KE of its COM. Consequently the acceleration of the COM would be greater if the ball slides down a frictionless slope without rolling than if rolls down a slope without slipping due to static friction, simply because all of its KE is the KE of its COM, all other things being equal.
For the above reasons, the COM of a ball sliding down a frictionless slope with reach the bottom sooner than the COM of a ball rolling without sliding on a slope with friction.
Hope this helps.
