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I am having a problem while dealing with the so called torque equation τ=Iα, which I am describing with the help of an illustration. Please help me out. Consider a rod of length L and mass ‘m’ lying on the positive x axis with its edge as origin. Suppose it has the z axis as the fixed axis of rotation. A force ‘F’ is applied on the COM of the rod, and parallel to the y axis. We can easily calculate the angular acceleration of the rod ‘α’ about the fixed axis of rotation by using τ=Iα. Now if we apply this relation to a point, other than the fixed axis of rotation, say the other end, what will be the new angular acceleration? I mean how can we find out angular acceleration of an object about a point other than the axis of rotation? How can we define angular acceleration about the end farther from the origin, when it is not rotating about that end?

Klyen Dave
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2 Answers2

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The idea is that the net torque about the axis of interest will tell you the net change in angular momentum about that axis (which in the case of rotation will tell you information about the angular speed).

In your first case (rotation about the origin), you considered only the force $F$ applied at the center of mass. But there is a fixed axis at the origin. You didn't consider any forces applied there. This is allowed because any forces applied there are coincident with the axis and result in zero torque. So only the applied force is necessary.

In your second case when you consider an axis at a different location, now forces applied at the origin may create a torque about the new axis. Unfortunately, you don't really have enough information to understand what they are. That means you can't easily calculate what the total change in angular momentum is around that axis.

Suppose the rod is rotating about the origin, how can we determine the angular speed or angular acceleration about the other end?

You can't really. In the simple case where the axis of interest coincides with the axis of rotation, then the angular momentum is proportional to the angular speed. So when you calculate change in angular momentum, the angular acceleration appears.

When the two axes do not coincide, then the relationship is no longer direct. You can calculate angular momentum about such an axis, but you can't really say that it has an angular speed about such a point.

BowlOfRed
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  • Suppose the rod is rotating about the origin, how can we determine the angular speed or angular acceleration about the other end? Shouldn’t it be necessarily defined about the origin only? Or is it so that in case of a fixed axis, angular acceleration, or torque about any other axis zero? – Klyen Dave Jul 03 '18 at 04:58
  • Now, assuming that the torque due to the hinge at origin is known, will the new angular acceleration that we get about the other end be the same as of that about the origin? – Klyen Dave Jul 03 '18 at 14:24
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Angular velocity is defined for a point with respect to some center point, which serves as a vertex for the angle to be measured.

If that center point is not fixed, we don't have an angle to measure.

So, you can measure the angular velocity (and acceleration) of any point on a rotating rod relative to a center point other than the origin of the rod, but this center point has to be fixed and therefore cannot lie on the rod.

V.F.
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