The gravitational PE of an object in the Earth's surface is given by, U = − G m M E r derive the expression for the force. Discuss the implication when r → ∞ and r → 0. Show your solutions step by step.

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The gravitational PE of an object in the Earth&#039;s surface is given by,  U  =  −  G            m              M        E              r  derive the expression for the force. Discuss the implication when   r  →  ∞ and   r  →  0. Show your solutions step by step.

Pradellalo · Accepted Answer

Newton&#039;s gravitation law states that each body of a certain mass attracts another body in the universe with a force (F) that is directly proportional to the product of their masses and is inversely proportional to the square of the distance (r) between them. This force is known as gravitational force. It is attractive in nature and always acts at the center of the line joining the two bodies. It is a conservative force which means that it can be written as the negative gradient of the gravitational potential energy (U) of the system.Write the given expression for the gravitational potential energy of a system.  U  =  −            G      m              M        g              r  Here, G is the universal gravitation constant.Write the gravitational force as a negative gradient of the gravitational potential energy.  F  =  −  ∇  U    F  =  −            d      U              d      r          N  o  w  ,    F  =  −      d          d      r        (            G      m              M        g              r    )    F  =  −  G  m      M    g        d          d      r        (      1    r    )    F  =  −  G  m      M    g    (            −      1              r      2        )    F  =            G      m              M        g                    r      2      Thus, the above-derived expression is the required expression for the gravitational force between two bodies.Write the above-derived expression for the gravitational force.  F  =            G      m              M        g                    r      2      When   r  →  ∞  F  ≃  0Hence, an object at an infinite distance from the surface of the earth will feel no gravitational force due to the earth.When   r  →  0,  F  →  ∞Hence, the gravitational force between an object and the earth is strongest when the object is at the center of the earth.

The gravitational PE of an object in the Earth's surface is given by, U=-G (mM_E)/(r) derive the expression for the force. Discuss the implication when r -> o and r -> 0.

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