# Newton's second law for rotation, when does it apply?

## Main Question or Discussion Point

I'm reading Halliday's chapter or rigid body rotation. In the derivation of Newton's second law for rotation, it is assumed that the object is hinged about some axis ( the connection would be a frictionless pin). The law is derived for such a connection, but is later applied to objects that aren't hinged. How is this?

To give an example, assume we have a uniform solid cylinder rolling on a horizontal surface. The normal force from the ground and the weight of cylinder both pass vertically through its center of mass. Friction is the only horizontal force, and there is no opposing horizontal "reaction" occurring at the center of mass of the disk. How can we apply the second law for rotation? There is no reaction at the axis of rotation.

How can a single force of friction cause rotation, without there being a couple?
Without the reaction, won't friction just cause translation instead of rotation?

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Doc Al
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I'm reading Halliday's chapter or rigid body rotation. In the derivation of Newton's second law for rotation, it is assumed that the object is hinged about some axis ( the connection would be a frictionless pin). The law is derived for such a connection, but is later applied to objects that aren't hinged. How is this?
You'll have to wait for your next level mechanics class for the more general derivation. (I don't recall how Halliday covers things; I'll check when I get the chance.) But that law is not restricted to objects with a fixed axis.

How can a single force of friction cause rotation, without there being a couple?
A couple is only needed for rotation without translation of the center of mass (since the net force would be zero). You can certainly have rotation with translation.

Without the reaction, won't friction just cause translation instead of rotation?
No. The friction does two things: It creates a translational acceleration of the center of mass and it creates a rotational acceleration about the center of mass. The resulting motion is a combination of translation and rotation.

A.T.
How can a single force of friction cause rotation, without there being a couple?
What do you think happens, if a stick floating in space is pushed perpendicularly at one end? Try it out with a falling stick.

You'll have to wait for your next level mechanics class for the more general derivation. (I don't recall how Halliday covers things; I'll check when I get the chance.) But that law is not restricted to objects with a fixed axis.

A couple is only needed for rotation without translation of the center of mass (since the net force would be zero). You can certainly have rotation with translation.
My own answer was the following: The assumption that the object is hinged to an axis is not the only assumption in halliday's derivation. It is also assumed that the coordinate system is attached to the axis of rotation. So If we wish to apply the second law for rotation (based on Halliday's hinge derivation) we would have to check for the presence of a "hinge reaction" in the coordinate system attached to the axis of rotation. If we find it, we can apply the equation.

Based on the above, If we seek such a coordinate system for the rolling cylinder, we would have to take the role of an observer moving with the center of mass of the cylinder. Such an observer would measure a fictitious force on the cylinder equal in magnitude to Macm, which should be applied to the center of mass of the cylinder. This force would have no effect on rotation since it passes through the axis of rotation (center of mass), but it will serve as the "hinge reaction" in our derivation of the second law for rotation.

In other words, this force, together with friction, would serve as the couple in this new coordinate system. So the conditions are exactly the same as the prerequisites assumed in deriving the second law (with hinge connection), therefore we can apply this law to the current situation. Makes sense?

I haven't read any advanced derivations yet so I would like to try to prove things based on Halliday's derivation for now.

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You can do this but you don't need to.
The reaction force is not necessary, as was already mentioned. And nor is a fixed axis necessary.
It's just that the things are simpler in this case.The author mentioned probably did not want to go to the general case, at that level. It's introductory physics.:)

CWatters
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To give an example, assume we have a uniform solid cylinder rolling on a horizontal surface. The normal force from the ground and the weight of cylinder both pass vertically through its center of mass. Friction is the only horizontal force, and there is no opposing horizontal "reaction" occurring at the center of mass of the disk.
Is it possible to have a friction force with "no opposing horizontal "reaction" occurring at the center of mass of the disk" ?

If the cylinder isn't accelerating then the horizontal forces must sum to zero. If it is accelerating then there must be a horizontal force at the centre of mass. no?

Doc Al
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Is it possible to have a friction force with "no opposing horizontal "reaction" occurring at the center of mass of the disk" ?
Why not?

If the cylinder isn't accelerating then the horizontal forces must sum to zero.
If the diagram is to be believed, the cylinder is accelerating. If it were rolling at constant speed, then there would be no friction. Here, whether intended or not, the cylinder must be slipping as it rolls.

I agree, though, that if the cylinder was meant to roll without slipping at constant velocity, then the force diagram is incorrect or incomplete. Perhaps a better example to analyze would have been a cylinder rolling without slipping down an incline.

If it is accelerating then there must be a horizontal force at the centre of mass. no?
No.

jbriggs444
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The mental picture that the diagram in the OP gives me is that of a cylinder on a rug with the rug being pulled out from under it rightwards. As others have indicated, this will impart both a rightward linear acceleration and a counter-clockwise torque.

Doc Al
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The mental picture that the diagram in the OP gives me is that of a cylinder on a rug with the rug being pulled out from under it rightwards.
I like it! No one said that the horizontal surface wasn't moving. :)

Is it possible to have a friction force with "no opposing horizontal "reaction" occurring at the center of mass of the disk" ?

If the cylinder isn't accelerating then the horizontal forces must sum to zero. If it is accelerating then there must be a horizontal force at the centre of mass. no?
First, we pick up a wheel. Then, we give it constant angular velocity (we spin it). THEN, we put it on a horizontal surface. Up to now, despite having a spin, the wheel has zero horizontal velocity relative to us or the ground; its center of mass has been at rest. The moment it touches the ground however, it starts moving forwards, right? If you're holding a spinning wheel in your hand and you put it on the ground what is gonna happen? Its gonna start moving forwards, and it does so from rest. So it gains linear acceleration and we conclude that there must be a forward friction force exerted on the wheel by the ground responsible for this acceleration.

Technically, this forward force is brought about as a response or reaction to the force exerted on the ground by the wheel. You can visualize this by thinking of the wheel as being bladed. The bladed wheel is spinning in the air, with the blades rotating counter clockwise. The ground is at rest. As soon as the wheel comes into contact with the ground, the blades start pushing into the ground, to the left (counter clockwise). Friction results as a reaction to this push.

And yes, there is slipping, since there is relative motion between the point of contact and the ground.

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