Explanation of Gravitational Potential Energy

Explanation of Gravitational Potential Energy

We had discussed earlier the notion of potential energy as being the energy stored in the body at its given position. If the position of the particle changes on account of forces acting on it, then the change in its potential energy is just the amount of work done on the body by the force. As we had discussed earlier, forces for which the work done is independent of the path are the conservative forces.

The force of gravity is a conservative force and we can calculate the potential energy of a body arising out of this force, called the gravitational potential energy. Consider points close to the surface of earth, at distances from the surface much smaller than the radius of the earth. In such cases, the force of gravity is practically a constant equal to mg, directed towards the centre of the earth. If we consider a point at a height h₁ from the surface of the earth and another point vertically above it at a height h₂ from the surface, the work done in lifting the particle of mass m from the first to the second position is denote by W₁₂

W₁₂ = Force × displacement

= mg (h₂ – h₁)                         (8.20)

If we associate a potential energy W(h) at a point at a height h above the surface such that 

W(h) = mgh + W_o                      (8.21)

(where W_o = constant) ; then it is clear that

W₁₂ = W(h₂) – W(h₁)                (8.22)

The work done in moving the particle is just

the difference of potential energy between its final and initial positions.Observe that the constant W_o cancels out in Eq. (8.22). Setting h = 0 in the last equation, we get   W ( h = 0 ) =W_o . h = 0 means points on the surface of the earth. Thus, W_o is the potential energy on the surface of the earth.

If we consider points at arbitrary distance from the surface of the earth, the result just derived is not valid since the assumption that the gravitational force mg is a constant is no longer valid. However, from our discussion we know that a point outside the earth, the force of gravitation on a particle directed towards the centre of the earth is

 \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9maalaaabaGaam4raiaad2eadaWgaaWcbaGaamyraaqabaGccaWG % TbaabaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaaaa!3D47! F = \frac{{G{M_E}m}}{{{r^2}}}\)                                (8.23)

where M_E = mass of earth, m = mass of the particle and r its distance from the centre of the earth. If we now calculate the work done in lifting a particle from r = r₁ to r = r (r₂ > r₁ ) along a vertical path, we get instead of Eq. (8.20)

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGxb % WaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9maapehabaWaaSaa % aeaacaWGhbGaamytaiaad2gaaeaacaWGYbWaaWbaaSqabeaacaaIYa % aaaaaaaeaacaWGYbWaaSbaaWqaaiaaigdaaeqaaaWcbaGaamOCamaa % BaaameaacaaIYaaabeaaa0Gaey4kIipakiaadsgacaWGYbaabaGaey % ypa0JaeyOeI0Iaam4raiaad2eadaWgaaWcbaGaamyraaqabaGccaWG % TbWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGYbWaaSbaaSqaaiaaik % daaeqaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadkhadaWgaaWc % baGaaGymaaqabaaaaaGccaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8 % UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 % caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daqadaqaaiaaiI % dacaGGUaGaaGOmaiaaisdaaiaawIcacaGLPaaaaaaa!7230! \begin{array}{l} {W_{12}} = \int\limits_{{r_1}}^{{r_2}} {\frac{{GMm}}{{{r^2}}}} dr\\ = - G{M_E}m\left( {\frac{1}{{{r_2}}} - \frac{1}{{{r_1}}}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {8.24} \right) \end{array}\)

In place of Eq. (8.21), we can thus associate a potential energy W(r) at a distance r, such that

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaabm % aabaGaamOCaaGaayjkaiaawMcaaiabg2da9iabgkHiTmaalaaabaGa % am4raiaad2eadaWgaaWcbaGaamyraaqabaGccaWGTbaabaGaamOCaa % aacqGHRaWkcaWGxbWaaSbaaSqaaiaaigdaaeqaaaaa!4281! W\left( r \right) = - \frac{{G{M_E}m}}{r} + {W_1}\),               (8.25)

valid for r > R ,

so that once again W ₁₂ = W(r₂) – W(r₁). Setting r = infinity in the last equation, W ( r = infinity ) = W₁ . Thus, W₁ is the potential energy at infinity. One should note that only the difference of potential energy between two points has a definite meaning from Eqs. (8.22) and (8.24). One conventionally sets W1 equal to zero, so that the potential energy at a point is just the amount of work done in displacing the particle from infinity to that point.

 Hence

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaabm % aabaGaamOCaaGaayjkaiaawMcaaiabg2da9iabgkHiTiaaisdadaWc % aaqaaiaadEeacaWGTbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamiBaa % aacqGHsislcaaIYaWaaSaaaeaacaWGhbGaamyBamaaCaaaleqabaGa % aGOmaaaaaOqaamaakaaabaGaaGOmaaWcbeaakiaadYgaaaaaaa!45F1! W\left( r \right) = - 4\frac{{G{m^2}}}{l} - 2\frac{{G{m^2}}}{{\sqrt 2 l}}\)

              fig.8.9.

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey % OeI0YaaSaaaeaacaaIYaGaam4raiaad2gadaahaaWcbeqaaiaaikda % aaaakeaacaWGSbaaamaabmaabaGaaGOmaiabgUcaRmaalaaabaGaaG % ymaaqaamaakaaabaGaaGOmaaWcbeaaaaaakiaawIcacaGLPaaacqGH % 9aqpcqGHsislcaaI1aGaaiOlaiaaisdacaaIXaWaaSaaaeaacaWGhb % GaamyBamaaCaaaleqabaGaaGOmaaaaaOqaaiaadYgaaaaaaa!49B9! = - \frac{{2G{m^2}}}{l}\left( {2 + \frac{1}{{\sqrt 2 }}} \right) = - 5.41\frac{{G{m^2}}}{l}\)

The gravitational potential at the centre of the square \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca % WGYbGaeyypa0ZaaOaaaeaacaaIYaaaleqaaOGaamiBaiaac+cacaaI % YaaacaGLOaGaayzkaaaaaa!3CBD! \left( {r = \sqrt 2 l/2} \right)\) is \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaabm % aabaGaamOCaaGaayjkaiaawMcaaiabg2da9iabgkHiTiaaisdadaGc % aaqaaiaaikdaaSqabaGcdaWcaaqaaiaadEeacaWGTbaabaGaamiBaa % aaaaa!3FA1! U\left( r \right) = - 4\sqrt 2 \frac{{Gm}}{l}\)