# Interpreting direction of torque

The direction of torque is along axis of rotation. What does this actually mean? Suppose a body is rotating anticlockwise and along its axis we placed someting. Would the material placed on axis experience a torque effect or force along upward direction as body is rotating anticlockwise? Or is there no effect on its state whatsoever?

• Is your first sentence a quote? if yes, then place it under quotation marks and denote where it is from. Also, please clarify the situation with a sketch or something. How is the "up" direction in relation to the rotation axis? What do you mean by "placing something on the body?" If the torque due to motion a result of friction or some other bonding mechanism? Jun 27, 2018 at 22:40

Given torque, what is the direction of motion? - The answers for this question may give you an answer to the first part of your question. The direction of torque only gives an idea about the axis (whether rotation is clockwise or anticlockwise) and is not the direction in which the force causes rotation of the body.

Mathematically, torque is the vector cross product of the force and the distance (from axis to the object) vectors. For a body placed on the axis, the distance from axis is zero and hence there is no torque acting on it.

• What I am thinking of is celing fan.As it rotates the air moves down as it were pulled from center where axis is located.Is my thinking wrong? Jun 27, 2018 at 19:55
• Ceiling fan blades can be angled either way so the same direction of spin, and torque, can either push the air up or down. I seem to recall there is even a design of fan that allows the blades to be angled as desired.
– user93146
Jun 27, 2018 at 20:25

Your question isn't very clear. But I feel there is some confusion as to what the equations of motion are, and what they mean.

First some definitions:

1. The linear velocity vector at a point is typically a result of the rotation about an axis not through the point. The amount of velocity gives us an idea of how far is the center of mass from the axis of rotation. The direction of the velocity vector is perpendicular to the direction of rotation. You can use the right hand rule to find which way something is moving when it is rotating about an axis.

$$\boldsymbol{v} = \boldsymbol{r} \times \boldsymbol{\omega}$$

2. Similarly the torque at a location is a result of a force being applied along a line (the line of action) not through that point. The direction of torque vector is perpendicular to the line of action. You can use the right hand rule to find which way something twists when a force is applied along an axis.

$$\boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F}$$

3. Linear momentum vector is the mass times the velocity vector of the center of mass. Regardless of how an object is rotating or moving, it the only the motion of the center of mass that defines the momentum vector $$\boldsymbol{p} = m \, \boldsymbol{v}_C$$

4. Force vector if the rate of change of linear momentum. Regardless loading conditions and motions, the net force on an object relates only to the motion of its center of mass. The direction of force is the same as the direction of the linear acceleration of the center of mass.

$$\boldsymbol{F} = \frac{{\rm d} \boldsymbol{p}}{{\rm d}t} = m \,\boldsymbol{a}_C$$

5. Angular momentum vector about the center of mass is calculated from the rotational motion about the center of mass. This rotation is shared with the entire body. Regardless of the motion of the center of mass, it is only the rotation about the center of mass that defines angular momentum. If the rotation is only about a fixed axis, then the scalar factor between rotation and angular momentum is the mass moment of inertia. For general motion the mass moment of inertia is 3×3 matrix that depends on the orientation of the body

$$\boldsymbol{L}_C = \mathrm{I}_C \boldsymbol{\omega}$$

6. Torque vector about the center of mass is the rate of change of angular momentum about the center of mass. Torque only affects the rotation of a body. The direction of torque is generally not the direction angular acceleration because the mass moment of inertia is not a scalar value causing some complex motion.

$$\boldsymbol{\tau} = \frac{{\rm d} \boldsymbol{L}_C }{{\rm d}t } = \mathrm{I}_C \boldsymbol{\alpha} + \boldsymbol{\omega} \times \mathrm{I}_C \boldsymbol{\omega}$$

In summary, a torque causes a body to rotate about its center of mass and a force to translate the center of mass. Their directions indicate where the rotation axis is and where the line of action is.