Relation Between Torque and Angular Acceleration

Consider a mass m moving in a circle of radius r , acted on by a tangential force F_t as shown in Figure 8.2.

  

Figure 8.2: Torque and angular acceleration

Using Newton's second law to relate F_t to the tangential acceleration a_t = r$\alpha$ , where $\alpha$ is the angular acceleration:

F_t = ma_t = mr$$\alpha$$

and the fact that the torque about the center of rotation due to F_t is: $\tau$ = F_tr , we get:

$$\tau$$ = mr ²$$\alpha$$.

For a rotating rigid body made up of a collection of masses m₁,m₂.... the total torque about the axis of rotation is:
 

$$\tau_{{\:\rm total}}^{} \sum \tau_{i}^{}$$ =

$$\sum \alpha$$ = (9)

The second line above uses the fact that the angular acceleration of all points in a rigid body is the same, so that it can be taken outside the summation.

Definition: Moment of Inertia of a rigid body

The moment of inertia, I , of a rigid body gives a measure of the amount of resistance a body has to changing its state of rotational motion. Mathematically,

$$\sum$$ (10)

Note: The units of moment of inertia are kg $\cdot$ m ².

This allows us to rewrite Equation 8.9 as:

 

$$\tau_{{\:\rm total}}^{} \alpha$$ (11)

which is the rotational analogue of Newton's second law.

Note:

• The complete set of dynamical equations needed to describe the motion of a rigid body consists of the torque equation given above, plus Newton's Second Law applied to the center of mass of the object:

  $$\vec{F}_{{\:\rm total}}^{}$$ = m$$\vec{a}_{cm}^{}$$

  where $\vec{a}_{cm}^{}$ is the acceleration of the center of mass.
• The moment of inertia, like torque must be defined about a particular axis. It is different for different choices of axes.
• Extended objects can again be considered as a very large collection of much smaller masses glued together to which the definition of moment of inertia given above can be applied.
• Examples of Moments of Inertia of Extended Objects:
  uniform hoop: I = mr ²
  cylindrical shell I = mr ²
  long thin rod (about middle) I = ${1\over 12}$mL ²
  long thin rod (about one end) I = ${1\over3}$mL ²
  solid cylinder I = ${1\over 2}$mr ²
  solid sphere I = ${2\over5}$mr ²
• The moment of inertia depends on how the mass is distributed about the axis. For a given total mass, the moment of inertia is greater if more mass is farther from the axis than if the same mass is distributed closer to the axis.