Explanation of Potential Energy
The word potential suggests possibility or capacity for action. The term potential energy brings to one’s mind ‘stored’ energy. A stretched bow-string possesses potential energy.When it is released, the arrow flies off at a great speed.The earth’s crust is not uniform, but has discontinuities and dislocations that are called fault lines.These fault lines in the earth’s crust are like ‘compressed springs’. They possess a large amount of potential energy. An earthquake results when these fault lines readjust.Thus, potential energy is the ‘stored energy’ by virtue of the position or configuration of a body.The body left to itself releases this stored energy in the form of kinetic energy. Let us make our notion of potential energy more concrete.
The gravitational force on a ball of mass m is mg . g may be treated as a constant near the earth 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 \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa % aaleaacaWGfbaabeaakmaabmaabaGaamiAaiabgYda8iabgYda8iaa % dkfadaWgaaWcbaGaamyraaqabaaakiaawIcacaGLPaaaaaa!3E22! {R_E}\left( {h < < {R_E}} \right)\) so that we can ignore the variation of g near the earth’s surface*.In what follows we have taken the upward direction to be positive. Let us raise the ball up to a height h. The work done by the external agency against the gravitational force is mgh. This work gets stored as potential energy. 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)=mgh
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,
\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9iabgkHiTmaalaaabaGaamizaaqaaiaadsgacaWGObaaaiaadAfa % daqadaqaaiaadIgaaiaawIcacaGLPaaacqGH9aqpcqGHsislcaWGTb % Gaam4zaaaa!42A5! F = - \frac{d}{{dh}}V\left( h \right) = - mg\)
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,\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaCa % aaleqabaGaaGOmaaaakiabg2da9iaaikdacaWGNbGaamiAaaaa!3B7F! {v^2} = 2gh\)
This equation can be written as \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaacaWGTbGaamODamaaCaaaleqabaGaaGOmaaaa % kiabg2da9iaad2gacaWGNbGaamiAaaaa!3E2E! \frac{1}{2}m{v^2} = mgh\)
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.
Physically, the notion of potential energy is applicable only to the class of forces where work done against the force gets ‘stored up’ as energy. When external constraints are removed, it manifests itself as kinetic energy.Mathematically, (for simplicity, in one dimension) the potential energy V(x) is defined if the force F(x) can be written as \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iabgkHiTmaalaaabaGa % amizaiaadAfaaeaacaWGKbGaamiEaaaaaaa!3EF4! F\left( x \right) = - \frac{{dV}}{{dx}}\)
This implies that
\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaca % WGgbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqa % aiaadIhadaWgaaadbaGaamyAaaqabaaaleaacaWG4bWaaSbaaWqaai % aadAgaaeqaaaqdcqGHRiI8aOGaeyypa0JaeyOeI0Yaa8qCaeaacaWG % KbGaamOvaiabg2da9iaadAfadaWgaaWcbaGaamyAaaqabaGccqGHsi % slcaWGwbWaaSbaaSqaaiaadAgaaeqaaaqaaiaadAfadaWgaaadbaGa % amyAaaqabaaaleaacaWGwbGaamiBaiaadAgaa0Gaey4kIipaaaa!5289! \int\limits_{{x_i}}^{{x_f}} {F\left( x \right)dx} = - \int\limits_{{V_i}}^{Vlf} {dV = {V_i} - {V_f}} \)
The work done by a conservative force such as gravity depends on the initial and final positions only. In the previous chapter we have worked on examples dealing with inclined planes. 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 \(\sqrt{2gh}\) irrespective of the angle of inclination.Thus, at the bottom of the inclined plane it acquires a 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 non- conservative.
The dimensions of potential energy are [ML2T –2] 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 \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam % Ovaiabg2da9iabgkHiTiaadAeadaqadaqaaiaadIhaaiaawIcacaGL % PaaacqGHuoarcaWG4baaaa!3FE0! \Delta V = - F\left( x \right)\Delta x\).(6.9.)
In the example of the falling ball considered in this section we saw how potential energy was converted to kinetic energy. This hints at an important principle of conservation in mechanics, which we now proceed to examine.