0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 6/13)

The rule for angular acceleration

If the magnitude of the angular velocity increases in time, then the angular accelerationvector has the same direction as that of the angular velocity. If the magnitude of the angular velocitydecreases in time, then the angular acceleration vector has the opposite direction as that of theangular velocity.

Defining torque as a vector quantity

The magnitude of a torque is the product of two terms:

1. The length of a line that connects the axis of rotation to the point where the force acts.Refer to this line as r.
2. The component of the force, Ft, that is perpendicular to this line.

Define theta as the angle between the line and the force vector F (not the tangential component of the force vector but the force vector itself). Then themagnitude of the tangential force vector is given by

Ft = Fmag*sin(theta)

where

• Ft represents the magnitude of the tangential force vector that is perpendicular to the line r
• Fmag is the magnitude of the force vector
• theta is the angle between the line and the force vector

Define a vector R

Let R represent a vector that lies along the line from the axis of rotation to the point where the force acts with its tail at the axis of rotation. I will refer tothe magnitude of this vector as Rmag

Doing a little algebra, we can write

T = r*Ft, or

Tvec = Rmag*Fmag*sin(theta)

where

• Tvec represents the tangential force vector
• Rmag is the magnitude of the vector R
• Fmag is the magnitude of the vector F

The cross product

Why do I refer to Tvec as a vector in the above equation?

You learned in an earlier module that the cross product of two vectors A and B is given by

AxB = Amag*Bmag*sin(angle between A and B)

where

• AxB represents the cross product of the vectors named A and B
• Amag is the magnitude of vector A
• Bmag is the magnitude of vector B

The torque vector

Comparing the torque vector with the cross product , we determine that

Tvec = RxF

The direction of the torque vector

Recall from the earlier module that the direction of Tvec is perpendicular to both R and F and obeys the right-hand rule in terms of its absolute direction.

The relationship of torque and rotational inertia

Combining this with what we learned earlier about the relationship among torque, rotational inertia, and rotational acceleration, we can write

Tvec = I*A

A general equation for net torque

The equation shown in Figure 5 is a general equation for net torque. The net torque about an axis of rotation is equal to the product of the rotational inertia about that axis and the angular acceleration.