0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 5/13)

We confirmed earlier that the rotational analog of mass is the rotational inertia . In the case of a single mass,

I = m*r^2

Substitution yields

Ft = m*r*a, or

Ft = (I/r^2)*r*a, or

Ft = (I/r)*a

Multiplying both side by r yields

r*Ft = I*a

Similar to Newton's second law

The equation in Figure 2 looks a lot like Newton's second law for translational motion, which is often expressed as

Force = mass * acceleration

In this case, the rotational inertia, I, is analogous to mass and the rotational acceleration, a, is analogous to translational acceleration.

If this similarity holds, that must mean that the term on the left side is analogous to force in the tangential motion scenario.

Torque, rotational inertia, and angular acceleration

The term on the left , consisting of the product of the force and the distance from the point ofapplication of the force to the axis of rotation is commonly known as the torque. The common symbol for torque is the Greek letter tau, which I will replace withthe character T in equations in this module.

T = r*Ft = I*a , or

T = I*a

Torque produces angular acceleration

Thus, we have determined that a torque, which is the product of a tangential force and the distance from the application point of this force to the axis of rotation, produces an angular acceleration.

This is beginning to look a lot like Newton's second law for rotation. We will refine it some more later to improve the analogy. To do that, we will need to define torque and angular accelerationas vector quantities. We will accomplish that using the cross product of two vectors that you learned about in an earlier module.

The convention for positive rotation

If you imagine a rotating object from a viewpoint in which the rotation axis is perpendicular to the page, it is conventional to define a counter clockwiserotation as the positive direction of rotation. That is the convention that I will use in this module.

You learned about the right-hand rule involving vectors in an earlier module. There is a similar right-hand rule that we use to describe rotating objects.

The right-hand rule for angular velocity

If you curl the fingers of your right hand in the direction of rotation of the object, your thumb willpoint in the direction of the angular velocity vector of the object.

In other words, if some object is spinning in the counter-clockwise direction inthe x-y plane, curling the fingers of your right hand in this direction results in your thumbpointing in the +z direction which we define to be the direction of the angular velocity vector.