Is Torque Equal on All Axes When a Rigid Body's Net Force Is Zero?

[miss photon]

Homework Statement

prove that when the net force on a rigid body is zero, the torque about any line perpendicular to the plane of the forces is equal.

Homework Equations

$$\tau$$=I$$\alpha$$

The Attempt at a Solution

it is known that angular velocity, and hence angular acceleration about any line is the same for a given rotating rigid body.
implies, $$\alpha$$ is same about all lines.
if we accept the above statement to be true, then I comes out to be equal about every axis, which is a contradiction.

 

[Feldoh]

If it's perpendicular the the cross-product will just be tau = |r||F|

 

[ogb p]

Therefore, we cannot assume that \alpha is the same about all lines.
Instead, we can prove that when the net force on a rigid body is zero, the torque about any line perpendicular to the plane of the forces is equal.

First, let's define what a rigid body is. A rigid body is a body that maintains its shape and size even when subjected to external forces. This means that the distances between any two points on the body remain constant.

Now, let's consider a rigid body that is subjected to a net force of zero. This means that the body is in a state of equilibrium, where the forces acting on it cancel each other out.

In this state, the body will not have any translational motion, but it may have rotational motion. This is because even though the net force is zero, there may be forces acting on different points of the body that create a torque.

The torque (\tau) on a rigid body is given by the formula \tau=I\alpha, where I is the moment of inertia of the body and \alpha is the angular acceleration.

Since the net force on the body is zero, the total torque on the body must also be zero. This means that the sum of all the torques acting on the body must be equal to zero.

Now, let's consider a line perpendicular to the plane of the forces. This line will pass through the center of mass of the body. Since the net force is zero, there will be no translational motion and the center of mass will remain at rest.

Therefore, the angular acceleration of the body about this line will also be zero. This means that the moment of inertia (I) about this line must also be zero.

Now, let's consider another line perpendicular to the plane of the forces, but passing through a different point on the body. Since the body is rigid, the distance between the two points will not change. This means that the moment of inertia about this line will also be equal to zero.

Hence, we can conclude that when the net force on a rigid body is zero, the torque about any line perpendicular to the plane of the forces is equal. This is because the moment of inertia about any line perpendicular to the plane of the forces will also be equal to zero, resulting in equal torques.

In conclusion, when the net force on a rigid body is zero, the torque about any line perpendicular