Are *two* external forces always required to cause rotation?

[qpwo]

As the title says, can there be any rotation if only one force is applied to an object at rest. For instance, if I had a rod laying flat on a frictionless surface, and I pushed one end, am I correct to say that Newton's second law says that all of that force goes into acceleration the center of mass, and none into rotation? Whereas, if I fixed the center (or applied another force), the rod would rotate?

Then, if this is correct, is there an easy way to figure out the center of rotation? For example, if the rod in the previous example were 10 cm long and lying along the x-axis (centered at x=0), and I pushed with 15 N at x=5cm, and 5 N at x=3cm, I would expect the rod to both accelerate and rotate, but I am not sure how I would find the center of rotation. It seems like the rotation would not be centered on the COM, but it almost seems like it woulld have to be, in order to keep the center of mass uniformly accelerating...

Thanks for the help!

 

[jbriggs444]

Try it yourself. Put a pencil down on your desktop. Bump it on the side near one end. Does it rotate? Does it translate?

 

[Nugatory]

No, not correct. A single force applied off-center to the center of mass will cause the object to rotate around its center of mass and cause the center of mass to accelerate. Two forces applied at different points will always cause rotation about the center of mass no matter where they are applied, and will also cause the center of mass to accelerate if the forces are not of equal magnitude and opposite direction so they cancel.

It's hard to form a good intuitive picture of these situations because it's really very tricky applying two constant forces to an object moving freely on a frictionless surface. The best way to imagine applying a constant force is to imagine pushing a coil spring into the object in such a way that the compression of the spring remains constant... But because the object is free to move on the frictionless surface it is accelerating away from you and you have to chase after it to pushing on the spring... and you have two springs for two forces, and their ends are moving in different directions at different speeds.

 

[qpwo]

No, not correct. A single force applied off-center to the center of mass will cause the object to rotate around its center of mass and cause the center of mass to accelerate. Two forces applied at different points will always cause rotation about the center of mass no matter where they are applied, and will also cause the center of mass to accelerate if the forces are not of equal magnitude and opposite direction so they cancel.

It's hard to form a good intuitive picture of these situations because it's really very tricky applying two constant forces to an object moving freely on a frictionless surface. The best way to imagine applying a constant force is to imagine pushing a coil spring into the object in such a way that the compression of the spring remains constant... But because the object is free to move on the frictionless surface it is accelerating away from you and you have to chase after it to pushing on the spring... and you have two springs for two forces, and their ends are moving in different directions at different speeds.

This mostly makes sense to me from your description. However, one question lingers. You say that two forces applied at different points will always cause rotation about the COM. How is it possible to get rotations about a different axis then? Isn't applying forces the only thing we can ever really do?

 

[mfb]

You say that two forces applied at different points will always cause rotation about the COM.

They don't have to, but they can (and in general, they do).

 

[Nugatory]

They don't have to, but they can (and in general, they do).

Ah - right. There are some special cases where you won't get any rotation, such as if the forces are equal in magnitude and applied in the same direction at equal distances on opposite sides of the center of mass.

 

[Nugatory]

You say that two forces applied at different points will always cause rotation about the COM. How is it possible to get rotations about a different axis then? Isn't applying forces the only thing we can ever really do?

You can put a pivot somewhere away from the center of mass. Of course, that's a different problem than an object moving freely on a frictionless surface. Also, in a three-dimensional situation (an object floating free in outer space, for example) there's more than one axis that passes through the center of mass, and which one the object rotates about will depend on the directions of the applied forces.

 

[mfb]

They don't have to be that special. They have to satisfy $F_1 \times r_1 + F_2 \times r_2 = 0$ where distances r are relative to the center of mass. In particular, you can fix three of those parameters and find a fourth that will give no rotation (unless you fix something to zero).

 

[qpwo]

You can put a pivot somewhere away from the center of mass. Of course, that's a different problem than an object moving freely on a frictionless surface. Also, in a three-dimensional situation (an object floating free in outer space, for example) there's more than one axis that passes through the center of mass, and which one the object rotates about will depend on the directions of the applied forces.

I guess I don't really see the difference between a pivot and a force that is exactly equal in magnitude (but opposite in direction from) the applied force...

 

[Nugatory]

I guess I don't really see the difference between a pivot and a force that is exactly equal in magnitude (but opposite in direction from) the applied force...

The force from a pivot varies in both magnitude and direction. At any moment, it's whatever adds vectorially to the applied force to produce a net force along a line between the center of mass and the pivot.