# B Torque Equation about a point other than point of rotation

1. Mar 23, 2017

### ovais

Hello all,

These days I am studying rotation and rolling of bodies. We know a body can be made to rotate about any point. Let's assume that an external force is acting on the rotating body fixed about the point of rotation, and this force made the body rotates faster that is its torque produces an angular acceleration in the body. And the calculation of angular acceleration(about the point of rotation) is easy to calculate using torque equation which says torque(in magnitude) equals magnitude of moment of inertia(about point of rotation) times angular acceleration of body about point of rotation (in magnitude).

I wonder if the torque equation work if I choose to quantify torque about some (new) point other than the point about which the body is actually rotating and plug it in, to find the angular acceleration about that new point? I mean do torque equation remains same about any general point or in Physics do we always need to use torque equation only about the point about which the body is actually rotating?

Regards
Thanks a bunch :)

2. Mar 23, 2017

### sophiecentaur

What you need is the Parallel Axis Theorem to help you (here) it will tell you the Moment of Inertia about any axis parallel to the one you start with through the CM. Then you find the value of torque round that axis and Bob's your Uncle.

3. Mar 23, 2017

### ovais

I have no problem in using in parallel axis theorm, however what I fear is that taking torque( F* perpendicular distance) and angular acceleration about a point other than the axis of rotation, would make equation inconsistent, since(as what I am thinking) the angular acceleration of the different points of the body will not come same and uniform. I am saying this based on the basic fundamentals which I study in the beginning of the lesson which says all points of the body will have the same angular velocity and same angular acceleration (if it exist)about the axis of rotation but angular velocity and angular acceleration of different points of the body will have different values if they are measured not about the actual point (or axis) of rotation.
So their is problem: in torque equation which points angular acceleration should we use as the angular acceleration the different points is different.

Thanks

4. Mar 23, 2017

### zwierz

Angular acceleration of the body as well as its angular velocity do not depend on a point of the body. There are two commonly used versions of
torque equation. Let a point $A$ be either a fixed point of the body or its center of mass then
$$J_A\dot{\boldsymbol\omega}+\boldsymbol\omega\times J_A\boldsymbol \omega=\boldsymbol M_A\qquad (*)$$
here $J_A$ is the inertia tensor about the point $A$ and $\boldsymbol M_A$ is the torque about the point $A$. Formula (*) does not hold for arbitrary point $A$

Last edited: Mar 23, 2017
5. Mar 23, 2017

### sophiecentaur

If there's rolling then you need to consider the linear momentum too. But I think that just requires another equation.