# Rotation about a fixed axis

"If a body is rotating about a fixed axis passing through its centre of mass, then net force on the body is zero. In this situation it is possible that angular velocity of body is variable." - why? This was written on my book...i don't get how is this a "factual" statement as there are sooo many physical scenarios where a continuous body is rotating about a fixed axis...I get it that the body will not be able to translate as COM is fixed (excluding vertical translation) but that doesn't mean net force has to be zero since it still CAN ROTATE..

If you sum over all the forces experienced by every "particle" of the object, it's the same as if you would take the force on the center of mass:

$$\vec{P}=\sum_{\alpha}m_{\alpha}\vec{\dot{r}}_{\alpha}=M\frac{m_1\vec{\dot{r}}_1+m_2\vec{\dot{r}}_2+m_3\vec{\dot{r}}_3+...}{m_1+m_2+m_3+...}=M\vec{\dot{R}}$$

where $$M$$ is the total mass, $$\vec{P}$$ is the total momentum, $$\vec{R}\equiv \frac{m_1\vec{r}_1+m_2\vec{r}_2+m_3\vec{r}_3+...}{m_1+m_2+m_3+...}$$ is the position of the center of mass, $$\alpha$$ is the index of a particle, $$\vec{r}_\alpha$$ is the position of a particle of index $$\alpha$$ and the dot means derivative with respect to time. From this follows that

$$\vec{F}=\sum_{\alpha}\vec{F}_{\alpha}=\frac{d}{dt}\sum_{\alpha}m_\alpha \vec{\dot{r}}_{\alpha}=\frac{d}{dt}M\vec{\dot{R}}=M\vec{A}$$

Where $$A$$ is the acceleration of the center of mass. This means, if the center of mass is not accelerating, then the net force on an object is $$0$$ and vice versa. Thus, if the object is rotating with respect to a fixed center of mass, the net force on it is $$0$$.

On the other hand, every individual part of the body will still experience a force towards the center (centripetal acceleration) if the object is rotating, but the sum of these forces will still be $$0$$ if the center of mass is stationary. A body can also experience a net torque, and the net force will still be $$0$$.

"If a body is rotating about a fixed axis passing through its centre of mass, then net force on the body is zero. In this situation it is possible that angular velocity of body is variable." - why? This was written on my book...

The net force is not necessarily zero unless the center of mass (COM) is fixed or moving at constant velocity. The angular velocity is variable (i.e., there is angular acceleration) only if there is a net torque acting about the COM. Otherwise the angular velocity is constant. Perhaps you should give the complete context of the book statement.

...i don't get how is this a "factual" statement as there are sooo many physical scenarios where a continuous body is rotating about a fixed axis...I get it that the body will not be able to translate as COM is fixed (excluding vertical translation) but that doesn't mean net force has to be zero since it still CAN ROTATE..

You are correct that there are a number of possible scenarios with different net torques and net forces.

1. If the body is rotating at constant angular velocity and its COM is either "fixed" or translating at constant velocity, then both the net torque and net force acting on the body is zero.

2. If the body is rotating with varying angular velocity (i.e., it has angular acceleration) and its COM is fixed or translating at constant velocity, then there is a net torque acting on the body but no net force.

3. If the body is rotating with varying angular velocity (i.e., it has angular acceleration) and its COM is accelerating, then there is both a net torque and net force acting on the body.

The quoted statement from the book can apply to either scenario 1 or 2 depending on whether or not the angular velocity is "variable". That, in turn, depends on whether or not there is a net torque acting on the body.

Hope this helps.

The net forces must be zero because, were they to be anything other than zero, it would cause the object to translate. The motion of its center of mass would be governed by F=ma.

Rotation being involved does not change this. A spinning object is still governed by F=ma. However, once we consider rotation of an object, we can consider two equal but opposite forces being applied to the object. From a F=ma perspective, the net force is still zero. But from a rotational perspective, those forces may be applied to different parts of the body, applying a torque.

You mention there being "sooo many physical scenarios where a continuous body is rotating about a fixed axis." Try to come up with one scenario where you can apply a non-zero net force to the rotating body without inducing a translation. If you come up with one, make sure you didn't ignore any forces (if there's an axel, it's easy to forget that that axel can apply a force). If you can come up with any examples where you feel this rule shouldn't hold, post it as a question and we can disect that particular scenario.

• Net force relates to translation.

• Net torque (a.k.a moment of force) relates to rotation.

You can see that directly from the laws governing how these properties cause changes in motion. Which is Newton's 2nd law of motion, in the usual (linear) and rotational case:

$$\sum F=ma \qquad\text{and}\qquad \sum \tau =I\alpha.$$

Nothing in these laws tell that net force interferes with rotation. A net force does not cause angular acceleration $$\alpha$$, which is change in spinning speed. Only a net torque does.