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We know that if there is a flat surface with friction, a ball rolling without slipping will conserve its energy, as friction does no work on the ball.

$$\Delta E = 0 $$

Because kinetic energy is conserved, then the translational kinetic energy will remain constant, meaning that the velocity of the center of mass of the ball will remain constant.

However, because there is a force of friction being applied, we know the following:

$$F_f=F_n \mu_s = mg\mu_s$$

$$a_{cm}=\frac{mg\mu_s}{m} $$

Since we know that this is the acceleration of the center of mass, then how does the center of mass of the ball continue forward with a constant velocity, as shown by conservation of energy?

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  • $\begingroup$ $F_f \le \mu_s N$, and $=0$ in this case. $\endgroup$
    – ytlu
    Commented Mar 11, 2021 at 17:37

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A ball that is rolling uniformly without slipping will have 0 friction force applied on it.

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  • $\begingroup$ Oh, so if it's moving at a constant velocity with a net force of 0, then it has no tendency to slip against the surface, meaning that there's no friction. Is this assumption correct? $\endgroup$
    – Arjun Inamdar
    Commented Mar 11, 2021 at 17:51
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    $\begingroup$ Basically, if $v\neq\omega R$, then the ball will slip until it is "rolling without slipping". Then you can make the floor frictionless for all it cares, since its motion doesn't change anymore. $\endgroup$
    – DanDan面
    Commented Mar 11, 2021 at 19:21
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    $\begingroup$ @ArjunInamdar There is no static or kinetic friction, but there will be energy loss due to internal friction associated with rolling resistance in the form of heat eventually slowing the ball down, as pointed out in my answer. $\endgroup$
    – Bob D
    Commented Mar 12, 2021 at 14:25
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You've made a classic error in problems involving friction - the equation $F_f = \mu mg$ describes the maximum possible force that friction can provide, not the force that it actually is providing. One can stand motionless on the floor and calculate $F_f$ from their weight and coefficient of friction between their shoes and the floor, but it's clear that friction isn't actually providing any force at all. Unless you have the condition that the object is "about to slip", the equation should actually be $F_f \leq \mu mg$. There's an upper limit on the force that friction can provide, but it can be less than that.

The friction equation computes the maximum value at which the ball could accelerate without slipping, but it doesn't say anything at all about whether the ball is actually accelerating faster, or slower, or not at all. In this case, the ball simply rolls with constant speed with no frictional force whatsoever, since it already has all the translational and rotational momentum it needs. The ball would roll exactly the same whether or not there is friction between it and the floor.

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We know that if there is a flat surface with friction, a ball rolling without slipping will conserve its energy, as friction does no work on the ball.

Static friction does not dissipate energy, and since the ball is not slipping there is no energy loss in the form of kinetic friction. However, the ball will lose energy and slow down due to rolling resistance.

Rolling resistance is the result of inelastic deformation of the material of a rolling object, such as the rubber of a tire, when it contacts the road. The material compresses when it contacts the road and decompresses when it leaves the road during each revolution. The squeezing and un squeezing of the material generates internal friction and heat. If the ball was a perfectly elastic body rolling resistance would not occur. However, real objects are not perfectly elastic.

For more details on rolling resistance, see this article from Wikipedia: https://en.wikipedia.org/wiki/Rolling_resistance.

Hope this helps.

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In these problems, note that static friction force $f$ is $$0\leq f \leq f_\mathrm{max}=\mu mg$$

The $f_\mathrm{max}=\mu mg$ corresponds to the maximum value of static friction.

In your case, $f=0$.

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The maximum static friction force $F = \mu N$ assures that the translational velocity of the ball can only increase or decreases if its angular velocity changes accordingly.

If there is a small and temporary slope downward for example, the component of the weight in the direction of the movement minus the friction force results in a temporary linear acceleration. An the resultant torque (because the forces are not colinear) increases the angular velocity. It is possible to find the friction force as a function of the external force, radius, mass and moment of inertia of the ball.

In a ideal case, with no external forces in the direction of the movement, the angular velocity and linear velocity keeps the same by inertia, and the friction force is zero.

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