Paradox of Rolling: Exploring the Conflicting Forces on an Ideal Wheel

[alphabeta1720]

When an ideal wheel rolls without slipping on a horizontal floor there is a frictional force exerted at the instantaneous point of contact between the wheel and the surface in the opposite direction to motion.

But this frictional force does no work on the moving wheel because the point of application of the force does not move.
Then Work-Energy Theorem implies change in Kinetic Energy $$\Delta$$ K = 0 because work done = 0.
But if analysed from Newton's Laws since there is an external force(friction) in the opposite direction to motion there must be an acceleration, which implies change in speed and hence change in kinetic energy.

Please correct me where am I wrong.
Thank you.

 

[rcgldr]

If an ideal wheel is rolling without slipping, then there is no friction force. On a real wheel, there is rolling resistance, energy is lost due to deformation at the contact patch. Even on a frictionless surface, rolling resistance would reduce the angular velocity of a rotating wheel (the linear velocity would not be affected on a frictionless surface).

 

[Hologram0110]

I don't see any contradiction. If an ideal wheel is rolling, there is no friction force. It is only when the wheel is trying to slip (accelerating, decelerating, working against a force) that there can be a friction force on the wheel. Otherwise, there are no two surfaces trying to slide against each other to cause friction.

Also, no work is done by the friction force because the friction force is perpendicular to the direction of motion of the point. This is because $$W=\int F\cdot dx$$.

 

[alphabeta1720]

If an ideal wheel is rolling without slipping, then there is no friction force. On a real wheel, there is rolling resistance, energy is lost due to deformation at the contact patch. Even on a frictionless surface, rolling resistance would reduce the angular velocity of a rotating wheel (the linear velocity would not be affected on a frictionless surface).

refer to Pg 265 of Physics by Resnick,Halliday,Krane. In it its written that there is frictional force.

 

[vaibhav1803]

If an ideal wheel is rolling without slipping, then there is no friction force. On a real wheel, there is rolling resistance, energy is lost due to deformation at the contact patch. Even on a frictionless surface, rolling resistance would reduce the angular velocity of a rotating wheel (the linear velocity would not be affected on a frictionless surface).

its not always true that friction is`zero in ideal rolling, but two things are implied here on saying "PURE ROLLING OF A RIGID BODY"
1. the Kinetic friction acting is zero
2. Static friction shall act at the sole point of contact having a finite value decided by the forces acting on the body.

Q.E.F. ..anything else you'd want to know.?

 

[HallsofIvy]

Also note that "force acting" does not necessarily imply work. If I apply a 100 N force to an object that does NOT move, there is no work done.

 

[Studiot]

R & H is not one of my favourite tomes, so I can't look at the example.

However it does make a difference why the wheel is turning.

What is true to say is that if there is no friction and no torque is applied to the wheel, then there is no turning moment and the wheel will not turn.

 

[D H]

refer to Pg 265 of Physics by Resnick,Halliday,Krane. In it its written that there is frictional force.

Citation needed, please. Are you sure you aren't talking about a discussion of an object rolling down a ramp without slipping? Note well: Even though there is a frictional force in the case of an case, it is not doing any work (in an ideal situation).

 

[rcgldr]

An object rolling down a ramp without slipping?

On an inclined plane, the total energy gain = m g h. The angular energy portion of the total energy is related to the friction force, which times the radius generates a torque, and that torque performs work by increasing the angular energy over time. It also opposes the component of force parallel to the inclined plane from gravity, reducing the rate of gain in linear speed, and the corresponding linear kinetic energy.

If the plane is horizontal, it's an ideal wheel, and it's already rolling without slipping, then it will continue to do so without any other forces, including any friction forces. Imagine the object and plane are in space and void of any external forces. If the object's angular velocity in radians per unit time, times it's radius, results in a surface speed equal and opposing to it's linear speed relative to the plane, it will appear to be rolling without slipping, even with no contact.

 

[Studiot]

When an ideal wheel rolls without slipping on a horizontal floor there is a frictional force exerted at the instantaneous point of contact between the wheel and the surface in the opposite direction to motion.

But this frictional force does no work on the moving wheel because the point of application of the force does not move.
Then Work-Energy Theorem implies change in kinetic energy K = 0 because work done = 0.
But if analysed from Newton's Laws since there is an external force(friction) in the opposite direction to motion there must be an acceleration, which implies change in speed and hence change in kinetic energy.

Please correct me where am I wrong.
Thank you.

Rolling does seem to cause more than its fair share of misconceptions.

Alphabeta did start off by saying 'on a horizontal surface'

He also specified an ideal wheel. This is one that does not distort out of round, like real ones, to sit on a small contact zone.

He is correct, as is R & H, in observing that there is tangential ( and therefore horizontal) friction force acting at the point of contact.
Yes this is the force given by the coefficient of static friction since an ideal wheel has no slippage.
He is further correct in stating that no work is done by this force.

The problem comes in applying Newton's laws.

The edge of the wheel is describing circular motion, with acceleration directed toward the centre (wheel hub).

Note it is the acceleration that is pependicular to the friction, not the displacement.

Since the frictional force is horizontal (tangential) it is perpendicular to the acceleration.

There is zero acceleration in the direction of the frictional force (the negative of the direction of horizontal translation of the wheel) so no work is done and Newton's laws are not at variance with the rest of the analysis.

 

[rcgldr]

R & H, in observing that there is tangential ( and therefore horizontal) friction force acting at the point of contact. Yes this is the force given by the coefficient of static friction since an ideal wheel has no slippage.

The coefficient of static friction times the normal force is the maximum amount of static friction between the object and the plane. The minimal amount is zero. Absent any other forces, the object will not experience angular or linear acceleration, so there is no net force acting on the object, and the friction force would be zero.

The only forces involved would be the downwards force of gravity and the equal and opposing upwards force from the plane, resulting in zero net force and no acceleration of the object.

 

[Studiot]

We haven't been told why(how) the wheel is turning.

 

[rcgldr]

We haven't been told why(how) the wheel is turning.

The original post stated the ideal wheel was already rolling. so the reason it's rolling is momentum (angular and linear). Even with zero friction, absent any other horizontal forces, it will continue to rotate and translate at the same speed.

 

[D H]

He is correct, as is R & H, in observing that there is tangential ( and therefore horizontal) friction force acting at the point of contact.

He is incorrect. An ideal wheel that is already rotating without slipping on a horizontal surface is subject to zero frictional force.

The problem comes in applying Newton's laws.

The edge of the wheel is describing circular motion, with acceleration directed toward the centre (wheel hub).

Note it is the acceleration that is pependicular to the friction, not the displacement.

So what? Work is the inner product of force and displacement,

$$W=\int_C \vec F \cdot d\vec x$$

Alternatively,

$$W=\int_C \vec F \cdot \vec v dt$$

An ideal wheel contacts the surface at exactly one point. When the wheel is rolling without slipping, the instantaneous velocity of that contact point with respect to the plane is identically zero. Zero velocity means zero work, regardless of the force being applied.

 

[D H]

We haven't been told why(how) the wheel is turning.

Irrelevant. The wheel is already turning.

Suppose instead it is not turning. A perfect bowling ball sent down a perfectly flat and perfectly horizontal lane, for example. In this case the ball initially will be slipping along the surface, and that will subject the ball to a frictional force. That will apply a torque to the ball, making it start to rotate. As the ball's rotation rate increases the frictional force will decrease, becoming identically zero at exactly the point where the ball is rotating without slipping.

 

[Redbelly98]

It's a common misconception to take the equation for static friction,

f_static,max = μ_s N​

and think that the static friction force is always equal to μ_sN. And while many textbook problems employ situations where that is true, in general f_static is simply less than or equal to f_static,max, and can in fact be zero.

 

[Andy Resnick]

When an ideal wheel rolls without slipping on a horizontal floor there is a frictional force exerted at the instantaneous point of contact between the wheel and the surface in the opposite direction to motion.

But this frictional force does no work on the moving wheel because the point of application of the force does not move.
Then Work-Energy Theorem implies change in Kinetic Energy $$\Delta$$ K = 0 because work done = 0.
But if analysed from Newton's Laws since there is an external force(friction) in the opposite direction to motion there must be an acceleration, which implies change in speed and hence change in kinetic energy.

Please correct me where am I wrong.
Thank you.

As others have correctly pointed out, there is no frictional force in this scenario.

http://www.usna.edu/Users/physics/mungan/Scholarship/RollingFriction.pdf

 

[Studiot]

As others have correctly pointed out, there is no frictional force in this scenario.

If there is no tangential force and no torque is applied to the centre of the wheel . there is no source of turning moment and it cannot turn.

It is turning because something is applying a torque or couple about its centre or hub.

 

[D H]

The wheel is already turning, Studiot. That is a given; the wheel was stated to be rolling without slipping. Since angular momentum is a conserved quantity, a torque is needed to change the already-rotating wheel's non-zero angular momentum.

 

[Studiot]

The wheel is already turning, Studiot. That is a given; the wheel was stated to be rolling without slipping. Since angular momentum is a conserved quantity, a torque is needed to change the already-rotating wheel's non-zero angular momentum.

Agreed.

But something had to start it turning, as with your bowling ball.

 

[Redbelly98]

Agreed.

But something had to start it turning, as with your bowling ball.

Yes, something or someone had to give it a push to start it moving, and the static friction force and net torque were nonzero at that time.

The discussion is really about what happens afterwards, when it is just rolling along at constant velocity. From the OP: "...an ideal wheel rolls without slipping on a horizontal floor..." This is like many other physics problems, such as "a block slides along a frictionless surface", where we're not concerned with what started the block moving in the first place.

 

[mechprog]

Do you expect rolling over ice?
Put it the other way, if friction is present do you expect the wheel to slow down (considering zero flattening)?
And one more thing, do you expect the friction to spontaneously start a wheel lying on a floor (is it sufficient and/or necessary to start the rolling?)?

 

[alphabeta1720]

So can we now say that after the wheel has started rolling there is no friction at the point of contact, and hence no acceleration and the wheel will continue to roll in the same manner?

 

[Redbelly98]

Yes. Again, that is for an ideal wheel.