4.5 Rotational Kinetics

Torque (t ) is the rotational analog to force. Torque is a force applied through a turning lever arm (see Figure 4.7).

In Figure 4.7, F represents the tangential component of the applied force, and R is the level arm. The represented object rotates with an angular velocity w in the counterclockwise direction. The axis of rotation is perpendicular to the picture’s plane. Torque, measured in Newtons (N), is equal to the product of the tangential component of the applied force and the length of the lever arm:

Now let us consider a rigid body composed of n elements. The mass of the body, M, is equal to the sum of the mass of each element:

Each element of mass, m_i, is located a distance r_i from the axis of rotation. The force applied to m_i is represented by F_i, and the angle between the extension of r_i and F_i is represented by j (phi). The torque applied to m_i, t _i, is equal to the tangential component of the applied force (F_i sin f _i) and r_i.

The tangential component of the applied force, F_i sin f _i, can also be represented by F_it. According to Newton’s Second Law,

(This is a restatement of F = ma.) Here, a_it can be replaced by r_ia , where a is the angular acceleration. Thus,

and the net torque, t , is equal to the sum of t_i for all i:

The quantity (sum from i to N of m_ir_i²) is called the moment of inertia (I), and is characteristic of a body. It depends on both the geometry of the body (the size and distribution of mass), and the location of the axis of rotation.

Let us now examine the moments of inertia of several common shapes. First, take the example of a hollow cyliner rotated around an axis perpendicular to its circular faces and equidistant from all points on the shell.

Since all points on the cylindrical shell are equidistant from the axis of rotation,

The moment of inertia of a such a rotating shell were it solid, however, would be

The moment of inertia of a spherical shell is

and the moment of inertia of a solid shell is