18.10 Work - kinetic energy theorem for rotational motion

There is one to one correspondence between quantities in pure translation and pure rotation. We can actually write rotational relations for work and kinetic energy by just inspecting corresponding relation of pure translational motion.

Beside similarity of relations, there are other similarity governing description of pure motions in translation and rotation. We must, first of all, realize that both pure motions are one-dimensional motions in description. It might sound a bit bizarre for rotational motion but if we look closely at the properties of vector attributes that describe pure rotational motion, then we would realize that these quantities are indeed one - dimensional.

Rotational quantities like angular displacement (θ), speed (ω) and accelerations (α) are one - dimensional quantities for rotation about a fixed axis. These vector attributes are either anticlockwise (positive) or clockwise (negative). If it is anticlockwise, then the corresponding rotational vector quantity is in the direction of axis; otherwise in the opposite direction.

Rotation

We observe yet another similarity between two pure motions. We have seen that the pure translation of a particle or a rigid body has unique (single value) motional attributes like position, displacement, velocity and acceleration. In the case of rotation, on the other hand, even if individual particles constituting the rigid body have different linear velocities or accelerations, but corresponding rotational attributes of the rotating body, like angular velocity and acceleration, have unique values for all particles.

Rotation

There are, as a matter of fact, more such similarities. We shall discuss them in the appropriate context as we further develop dynamics of pure rotational motion.

Expressions of work and energy

The expressions of work in pure translation in x- direction (one dimensional linear motion) is given by :

$$\begin{array}{l}đW=F_xđx\\ \Rightarrow W=\int F_xđx\end{array}$$

The corresponding expressions of work in pure rotation (one dimensional rotational motion) is given by :

Obviously, torque replaces component of force in the direction of displacement (Fx) and angular displacement (θ) replaces linear displacement (x). Similarly, the expression of kinetic energy as derived earlier in the course is :

Here, moment of inertia (I) replaces linear inertia (m) and angular speed (ω) replaces linear speed (v). We, however, do not generally define gravitational potential energy for rotation as there is no overall change in the position (elevation) of the COM of the body in pure rotation. This is particularly the case when axis of rotation passes through COM.