# Will a sphere rotate on a frictionless inclined surface?

In summary, the conversation discusses a scenario where a sphere and a ring roll down a friction-less inclined plane and the topic of torque and rotational acceleration is brought up. The student argues that, without friction, there will be no torque around the center of mass to produce a rotational acceleration. The teacher disagrees and says that the point of contact can be used as a pivot point. The student questions if the normal force passes through the center of mass of the sliding object and the conversation concludes with a recommendation for further study in advanced physics textbooks.
Well, my physics teacher taught us about rotation the other day and I came across a scenario where a sphere and a ring roll down a friction-less inclined plane from a point of absolute rest. I found it counter-intuitive as I started to think about why would they start rolling in the first place. There is no off-set force that can provide torque. So, I raised my hand and argued that the normal force and gravity pass through the center of mass of the body, so the sphere ought to just slide down the plane. Then he drew a diagram and a gravity vector going straight down the center of the sphere and said gravity provides torque if we consider the point of contact as the pivot. At that point I didn't know what else to say and pretended to be satisfied by that. But, I'm not and that's why I'm here. Please help!

You are correct and your teacher is wrong. If there is no friction, there will be no torque about the center of mass to produce a rotational acceleration.

Then he drew a diagram and a gravity vector going straight down the center of the sphere and said gravity provides torque if we consider the point of contact as the pivot.
His mistake is not realizing that the point of contact is accelerating and thus different rules apply.

PeroK
Doc Al said:
You are correct and your teacher is wrong. If there is no friction, there will be no torque about the center of mass to produce a rotational acceleration.His mistake is not realizing that the point of contact is accelerating and thus different rules apply.
Can you please elaborate as to what mathematical statement can be used as an evidence that the ball will definitely slide and not spin about it COM. His thinking doesn't really matter, as long as I understand what is happening.

He should realize and accept that you can always use the COM as your reference point. Since there is no torque about the COM, there will be no rotational acceleration. Simple as that.

If he chooses to use the point of contact as the reference point, that's fine, but a more advanced analysis is required: The bottom line is that a proper analysis using that accelerating reference point will also show that there will be no rotational acceleration about the COM. (This is covered in every Classical Mechanics textbook. It's not really that advanced, just one step above Intro physics.)

Doc Al said:
You are correct and your teacher is wrong. If there is no friction, there will be no torque about the center of mass to produce a rotational acceleration.
That was my first thought, but then I found I myself thinking about the normal force exerted by the surface of the inclined plane... Does it pass through the center of mass of the sliding object? If I were to try standing on a frictionless incline looking downhill, my feet would go out from under me and the back of my head would hit the surface.

Nugatory said:
That was my first thought, but then I found I myself thinking about the normal force exerted by the surface of the inclined plane... Does it pass through the center of mass of the sliding object?
For spheres and rings, yes. (That's what we're talking about here. I think!)

Doc Al said:
He should realize and accept that you can always use the COM as your reference point. Since there is no torque about the COM, there will be no rotational acceleration. Simple as that.

If he chooses to use the point of contact as the reference point, that's fine, but a more advanced analysis is required: The bottom line is that a proper analysis using that accelerating reference point will also show that there will be no rotational acceleration about the COM. (This is covered in every Classical Mechanics textbook. It's not really that advanced, just one step above Intro physics.)
Thank you for your time, my query feels answered. What textbooks would you recommend for one level above the introductory level, as I would like to step into physics at home.

Doc Al said:
For spheres and rings, yes. (That's what we're talking about here. I think!)
Ah - yes, of course. I posted, went off to do start doing the dishes, and then realized that we're talking about circular spheres and rings so a vector perpendicular to the surface will pass through the center. I suspect that OP's teacher made the same mistake, failing to recognize the the importance of the symmetry in this case.

Nugatory said:
Ah - yes, of course. I posted, went off to do start doing the dishes, and then realized that we're talking about circular spheres and rings so a vector perpendicular to the surface will pass through the center.

You can easily simulate the situation you proposed in your previous post by standing on roller skates on a ramp or hill. If you stay on your feet a line perpendicular to the ramp will pass through your feet and your center of mass, something that most roller skaters can manage, and then you'll move down the ramp without rotating. On the other hand, if you fell over instead, you would still slide down the frictionless ramp without rotating.

Then he drew a diagram and a gravity vector going straight down the center of the sphere and said gravity provides torque if we consider the point of contact as the pivot.
He is wrong as Doc Al wrote. Here is another way to think about it: A uniform gravitational field alone never makes stuff spin, only the non-uniformly applied contact forces do, if they create a torque around the COM.

Teacher is wrong as others have said.

Then he drew a diagram and a gravity vector going straight down the center of the sphere and said gravity provides torque if we consider the point of contact as the pivot.

Draw a free body diagram of the ball.

There is no friction so the reaction force between the ball and slope must be normal to the slope1. This means that the reaction force also passes through the COM of the ball. So both gravity and the reaction force pass through the COM.

1) It's useful to remember this for other situations such as ladders leaning against frictionless walls.

Just another take on this. Simply decompose the gravitational force into normal and tangential components. The normal component cancels the normal force from the slope, leaving simple tangential acceleration.

When there is friction, you have an additional force up the slope. Note that this not only provides the torque for rotation, but also reduces the tangential acceleration. In your teacher's analysis, therefore, the energy would not add up. The rotational energy would come with no reduction to linear kinetic energy.

Jordan Regan and Ibix
PeroK said:

Just another take on this. Simply decompose the gravitational force into normal and tangential components. The normal component cancels the normal force from the slope, leaving simple tangential acceleration.

When there is friction, you have an additional force up the slope. Note that this not only provides the torque for rotation, but also reduces the tangential acceleration. In your teacher's analysis, therefore, the energy would not add up. The rotational energy would come with no reduction to linear kinetic energy.
But, shouldn't that be the case anyway? That friction is necessary to BEGIN rolling but once you start rolling, the instantaneous velocity of contact point w.r.t. ground is zero. Hence friction should not cost any energy as it does ZERO work. Am I right?

But, shouldn't that be the case anyway? That friction is necessary to BEGIN rolling but once you start rolling, the instantaneous velocity of contact point w.r.t. ground is zero. Hence friction should not cost any energy as it does ZERO work. Am I right?
In the case of friction it does zero work. The total kinetic energy of the sphere, rotational plus linear, equals the loss of gravitational potential energy. Friction has the effect of causing some of the energy to be rotational - at the expense of linear kinetic energy.

This scenario is worth studying.

But, if a sphere starts to roll without friction, then there is nothing to reduce the linear acceleration. So, all the potential energy must go to linear kinetic energy and the rotational energy must come for free.

But, shouldn't that be the case anyway? That friction is necessary to BEGIN rolling but once you start rolling, the instantaneous velocity of contact point w.r.t. ground is zero. Hence friction should not cost any energy as it does ZERO work. Am I right?
Static friction dissipates no energy. Kinetic friction and rolling resistance do.

Late to the party here. Another way of seeing the error of the teacher's claim...

Yes, if one uses the point of contact as the [instantaneous] axis of rotation then the force of gravity amounts to an unbalanced torque. However, one must realize that not all angular momentum consists of rotation. In this case, the resulting angular momentum increase is accounted for by the linear acceleration of the sphere or ring.

When the center of mass of an object is moving on a line that does not pass through the chosen axis then that motion has non-zero angular momentum given by ##\vec{L}=\vec{r}\times\vec{p}## where L is the angular momentum, r is the offset of the center of mass from the axis of rotation and p is the linear momentum of the object. If the object accelerates, its linear momentum is increasing along with the associated angular momentum.

jbriggs444 said:
Late to the party here. Another way of seeing the error of the teacher's claim...

Yes, if one uses the point of contact as the [instantaneous] axis of rotation then the force of gravity amounts to an unbalanced torque. However, one must realize that not all angular momentum consists of rotation. In this case, the resulting angular momentum increase is accounted for by the linear acceleration of the sphere or ring.

When the center of mass of an object is moving on a line that does not pass through the chosen axis then that motion has non-zero angular momentum given by ##\vec{L}=\vec{r}\times\vec{p}## where L is the angular momentum, r is the offset of the center of mass from the axis of rotation and p is the linear momentum of the object. If the object accelerates, its linear momentum is increasing along with the associated angular momentum.

@adesh123 I think the above is the best general explanation. You could just as well drop the ball vertically and calculate a non-zero gravity torque around some random point off the vertical. That torque will not make the ball spin during the fall, even tough it will change the angular momentum around that point.

I still think drawing a free body diagram for the ball is the way to go. It's obvious from that there is no torque on the ball.

CWatters said:
I still think drawing a free body diagram for the ball is the way to go. It's obvious from that there is no torque on the ball.
Except, of course, for the fact that there is a torque on the ball about the point of contact.

jbriggs444 said:
Except, of course, for the fact that there is a torque on the ball about the point of contact.
Suppose I don't know if the ball will slide or spin, what mathematical piece of information would lead me to a conclusion? I'm more of an analytical person than imaginative. It is true that the angular momentum can be provided by linear motion, but what evidence states that it will slide. (P.S. I'm not being in denial or over-skeptical, just want to learn how to go about things in physics)

Suppose I don't know if the ball will slide or spin, what mathematical piece of information would lead me to a conclusion?
There is no torque about its center of mass (on which point I agree with @CWatters). So it will not spin.

More generally, it will slide if its linear acceleration exceeds its rotational acceleration multiplied by its radius. Since its rotational acceleration is zero and its linear acceleration is non-zero, this condition is trivially met. You would typically determine the two accelerations by drawing free body diagrams, determining relevant forces and solving the resulting equations.

Suppose I don't know if the ball will slide or spin, what mathematical piece of information would lead me to a conclusion? I'm more of an analytical person than imaginative. It is true that the angular momentum can be provided by linear motion, but what evidence states that it will slide. (P.S. I'm not being in denial or over-skeptical, just want to learn how to go about things in physics)
The key piece of information is the maximum friction force.

As I mentioned in an earlier post, you should study a sphere and/or a cylinder moving down a frictional slope.

jbriggs444 said:
Except, of course, for the fact that there is a torque on the ball about the point of contact.
A free body diagram of the ball has no "point of contact", just the ball and the forces acting on it. No?

CWatters said:
A free body diagram of the ball has no "point of contact", just the ball and the forces acting on it. No?
A free body diagram used for computing torque most certainly has the points where forces act and where the axis of rotation is assigned.

## 1. Will a sphere roll or slide on a frictionless inclined surface?

A sphere will roll on a frictionless inclined surface due to the presence of a torque caused by the force of gravity acting on its center of mass.

## 2. What is the relationship between the angle of inclination and the speed of the rolling sphere?

The speed of the rolling sphere is directly proportional to the angle of inclination. As the angle increases, the speed of the sphere also increases.

## 3. Does the mass of the sphere affect its ability to roll on a frictionless inclined surface?

The mass of the sphere does not affect its ability to roll on a frictionless inclined surface. As long as there is a torque acting on the sphere, it will roll regardless of its mass.

## 4. Can a sphere roll indefinitely on a frictionless inclined surface?

Yes, a sphere can roll indefinitely on a frictionless inclined surface as long as there is no external force acting on it to stop its motion.

## 5. How does the radius of the sphere affect its rolling motion on a frictionless inclined surface?

The radius of the sphere does not affect its rolling motion on a frictionless inclined surface. As long as there is a torque acting on the sphere, it will roll regardless of its radius.

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