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InterviewSolution
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Define gravitational potential energy and derive an expression for it associated with two particles of masses m_1" and "m_2. |
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Answer» Solution :Gravitational potential energy of a body at a point in a gravitational FIELD of another. It is defined as the amount of work done in brining the given body from infinity to that point in the field is called Gravitational potential energy. Expression for gravitational potential energy : CONSIDER two PARTICLES of masses `m_1" and "m_2` are placed at the POINTS 'O' and p respectively. Let the distance between the two particles is 'r' i.e.,OP=r.  Let us calculate the gravitational potential energy of the particle of mass `m_2` placed at point p in the gravitational filed of `m_1`. Join OP and extended it in forward direction. Consider two points A and B on this line such that OA= x and AB=dx. The gravitational force of attraction on the prticle at A, `F=(Gm_1m_2)/(x^2)` Small amount of work done in bringing the particle without accletation through a very small distance AB is, dW=Fdx `=(Gm_1m_2)/(x^2)dx`. Total workdone in bringing the particle from infinity to the point P is, `W=int_(oo)^(r )(Gm_1m_2)/(x^2)dx=Gm_1m_2 int_(oo)^(r )x^(-2)dx` `= -Gm_1m_2[(1)/(x)]_(oo)^(r )= -Gm_1m_2[(1)/(r )-(1)/(oo)] =(-Gm_1m_2)/(r )`. Since, this work done is stored in the particle its gravatational potential energy. Therefore, gravitational potential energy of the particle of mass `m_2` placed at point 'p' in the gravitationl field of particle of mass `m_1` at distance r is, `U=(-Gm_1m_2)/(r )` Here, negetive sign shows that the potential energy is due to ATTRACTIVE gravitational force between two particles. |
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