6.10 Various forms of energy : the law of conservation of energy

CONSERVATIVE AND NONCONSERVATIVE FORCES

If the two forces on the body consist of a conservative force \(F_{c}\) and a non-conservative force \(F_{n c}\), the conservation of mechanical energy formula will have to be modified. By the Work-Energy theorem
\(
\begin{array}{lc}
& \left(F_{c}+F_{n c}\right) \Delta x=\Delta K \\
\text { But } & F_{c} \Delta x=-\Delta V \\
\text { Hence, } & \Delta(K+V)=F_{n c} \Delta x \\
& \Delta E=F_{n c} \Delta x
\end{array}
\)
where \(E\) is the total mechanical energy. Over the path, this assumes the form
\(E_{f}-E_{i}=W_{n c}\)
where \(W_{n c}\) is the total work done by the hon-conservative forces over the path. Note that, unlike the conservative force, \(W_{n c}\) depends on the particular path \(i\) to \(f\).

Example 6.15:

Suppose a block of mass \(m\) rests on a rough horizontal table (Figure 6.7). It is dragged horizontally towards the right through a distance \(l\) and then back to its initial position. Let \(\mu\) be the friction coefficient between the block and the table. Calculate the work done by friction during the round trip.

Figure 6.7

Solution: The normal force between the table and the block is \(N=m g\) and hence the force of friction is \(\mu m g\). When the block moves towards the right, friction on it is towards the left and the work by friction is \((-\mu m g l)\). When the block moves towards the left, friction on it is towards the right and the work is again \((-\mu m g l)\).

Hence, the total work done by the force of friction in the round trip is \((-2 \mu m g l)\).

We divide the forces in two categories (a) conservative forces and (b) nonconservative forces. If the work done by a force during a round trip of a system is always zero, the force is said to be conservative. Otherwise, it is called nonconservative.
Conservative force can also be defined as follows :
If the work done by a force depends only on the initial and final states and not on the path taken, it is called a conservative force.

VARIOUS FORMS OF ENERGY: THE LAW OF CONSERVATION OF ENERGY

We have seen that it can be classified into two distinct categories: one based on motion, namely kinetic energy; the other on configuration (position), namely potential energy. Energy comes in many forms which transform into one another in ways that may not often be clear to us.

Heat
We have seen that the frictional force is not a conservative force. However, work is associated with the force of friction, As an example consider a block of mass \(m\) sliding on a rough horizontal surface with speed \(v_{0}\) comes to a halt over a distance \(x_{0}\). The work done by the force of kinetic friction \(f\) over \(x_{0}\) is \(-f x_{0}\). By the work-energy theorem \(m v_{o}^{2} / 2=f x_{0}\). On examination of the block and the table, we would detect a slight increase in their temperatures. The work done by friction is not ‘lost’, but is transferred as heat energy. A quantitative idea of the transfer of heat energy is obtained by noting that \(1 \mathrm{~kg}\) of water releases about \(42000 \mathrm{~J}\) of energy when it cools by \(10^{\circ} \mathrm{C}\). Similarly, other sources of energy are chemical, electrical, and nuclear.

The Equivalence of Mass and Energy

Till the end of the nineteenth century, physicists believed that in every physical and chemical process, the mass of an isolated system is conserved. Matter might change its phase, e.g. glacial ice could melt into a gushing stream, but matter is neither created nor destroyed; Albert Einstein (1879-1955) however, showed that mass and energy are equivalent and are related by the relation
\(E=m c^{2} \dots (6.11)\)
where \(c\), the speed of light in vacuum is approximately \(3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\). Thus, a staggering amount of energy is associated with a mere kilogram of matter
\(E=1 \times\left(3 \times 10^{8}\right)^{2} \mathrm{~J}=9 \times 10^{16} \mathrm{~J} .\)
This is equivalent to the annual electrical output of a large \((3000 \mathrm{MW})\) power generating station.

The Principle of Conservation of Energy

When all forms of energy are taken into account, we arrive at the generalized law of conservation of energy. Energy can never be created or destroyed, it can only be changed from one form into another.