13.1 Work and energy

Kinetic energy

One of the most common energy that we come across in our day to day life is the energy of motion. This energy is known as kinetic energy and defined for a particle of mass "m" and speed "v" as :

$$\begin{array}{l}K=\frac{1}{2}m{v}^2\end{array}$$

Kinetic energy arises due to "movement" of a particle. The main characteristics of kinetic energy are as follows :

• The expression of Kinetic energy involves scalar quantities mass "m" and speed "v". Importantly, it involves speed i.e. the magnitude of velocity - not vector velocity. Therefore, kinetic energy is a scalar quantity.
• Both mass "m" and speed "v" are positive scalar quantities. Therefore, kinetic energy is a positive scalar quantity. This means that a particle can not have negative kinetic energy.
• Kinetic energy of a particle, at rest, is zero.
• Greater the speed or mass, greater is the kinetic energy and vice-versa.

The SI unit of kinetic energy is $\mathrm{kg}-m^2{s}^{-2}$ , which is known as "Joule". Since kinetic energy is a form of energy, Joule (J) is SI unit of all types of energy.

Work and kinetic energy

We have reached the situation when we can attempt to relate work with energy (actually kinetic energy). The relationship is easy to visualize in terms of the motion of a body.

In order to fully appreciate the connection between work and kinetic energy, we consider an example. A force is applied on a block such that component of force is in the direction of the displacement as shown in the figure. Here, work by force on the block is positive. During the time force does positive work, the speed and consequently kinetic energy of the block increases ( $K_f>K_i$ ) as block moves ahead with certain acceleration. We shall know later that the kinetic energy of the block increases by the amount of work done by net external force on it. This is what is known as "work - kinetic energy" theorem and will be the subject matter of a separate module.

For the time being, we concentrate on work and determine its relationship with energy in qualitative term. Since kinetic increases by the amount of work done on the particle, it follows that work, itself, is an energy which can be added as kinetic energy to the block. The other qualification of the work is that it is the "energy" which is transferred by the force from the surrounding to the block. Clearly, force here is an agent, which does the work to increase the speed of the object and hence to increase its kinetic energy. In this sense, "work" by a force is the energy transferred "to" the block, on which force is applied.

We, now, consider the reverse situation as illustrated in the figure below. A force is applied to retard the motion of a block. Here, the component of force is in the opposite direction to the displacement. The work done on the block is negative. The kinetic energy of the block decreases ( $K_f<K_i$ ) by the amount of work done by the force. In this case, kinetic energy is transferred "out" of the block and is equal to the amount of negative work by force. Here, "work" by a force is the energy transferred "from" the block, on which force is applied. Thus, we can define "work" as :