6.7 The concept of potential energy

THE CONCEPT OF POTENTIAL ENERGY

Potential energy is the ‘stored energy’ by virtue of the position or configuration of an object or body. A stretched bow string possesses potential energy. The object or body left to itself releases this stored energy in the form of kinetic energy. 

The gravitational force on a ball of mass \(m\) is \(m g . g\) may be treated as a constant near the earth’s surface. By ‘near’ we imply that the height \(h\) of the ball above the earth’s surface is very small compared to the earth’s radius \(R_{E}\left(h<<R_{E}\right)\) so that we can ignore the variation of \(g\) near the earth’s surface. Let us raise the ball up to a height \(h\). The work done by the external agency against the gravitational force is \(m g h\). This work gets stored as potential energy. The gravitational potential energy of an object, as a function of the height \(h\), is denoted by \(V(h)\) and it is the negative of work done by the gravitational force in raising the object to that height.
\(
V(h)=m g h
\)
If \(h\) is taken as a variable, it is easily seen that the gravitational force \(F\) equals the negative of the derivative of \(V(h)\) with respect to \(h\). Thus,
\(F=-\frac{\mathrm{d}}{\mathrm{d} h} V(h)=-m g\)
The negative sign indicates that the gravitational force is downward. When released, the ball comes down with an increasing speed. Just before it hits the ground, its speed is given by the kinematic relation,
\(v^{2}=2 g h\)
This equation can be written as
\(\frac{1}{2} m v^{2}=m g h\)
which shows that the gravitational potential energy of the object at height \(h\), when the object is released, manifests itself as kinetic energy of the object on reaching the ground.

Mathematically, (for simplicity, in one dimension) the potential energy \(V(x)\) is defined if the force \(F(x)\) can be written as
\(F(x)=-\frac{\mathrm{d} V}{\mathrm{~d} x}\)
This implies that
\(\int_{x_{i}}^{x_{f}} F(x) \mathrm{d} x=-\int_{V_{i}}^{V_{f}} \mathrm{~d} V=V_{i}-V_{f}\)
The work done by a conservative force such as gravity depends on the initial and final positions only. If an object of mass \(m\) is released from rest, from the top of a smooth (frictionless) inclined plane of height \(h\), its speed at the bottom is \(\sqrt{2 g h}\) irrespective of the angle of inclination. Thus, at the bottom of the inclined plane, it acquires kinetic energy, mgh. If the work done or the kinetic energy did depend on other factors such as the velocity or the particular path taken by the object, the force would be called nonconservative.

The dimensions of potential energy are \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) and the unit is joule \((J)\), the same as kinetic energy or work. To reiterate, the change in potential energy, for a conservative force, \(\Delta V\) is equal to the negative of the work done by the force
\(\Delta V=-F(x) \Delta x \dots(6.9)\)
In this example of the falling ball considered in this section, we saw how potential energy was converted to kinetic energy.