Two bodies of mass m and 9m are placed at a distance of R. The gravitational potential on the line joining the bodies where the gravitational fied equals zero, will be (G = gravitational contant):
\(-\frac{20 Gm}{R}\)
\(-\frac{8Gm}{R}\)
\(-\frac{12Gm}{R}\)
\(-\frac{16Gm}{R}\)
The Correct Option is D
Solution and Explanation
position of neutral point(zero gravitational field)
\(r_1=\frac{\sqrt(m_1)R}{\sqrt{m_1+\sqrt{m+2}}}=\frac{\sqrt{m}R}{\sqrt m+\sqrt{9m}}=\frac{R}{4}\)
\(r_2=R-\frac{R}{4}=\frac{3R}{4}\)
Now gravitational potential at point P
\(V_p=-\frac{GM}{\frac{R}{4}}-\frac{9GM}{\frac{3R}{4}}\)
\(=-\frac{16GM}{R}\)
Therefore, the correct option is (D) : \(-\frac{16Gm}{R}\).
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Concepts Used:
Gravitational Potential Energy
The work which a body needs to do, against the force of gravity, in order to bring that body into a particular space is called Gravitational potential energy. The stored is the result of the gravitational attraction of the Earth for the object. The GPE of the massive ball of a demolition machine depends on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between GPE and the mass of an object. More massive objects have greater GPE. Also, there is a direct relation between GPE and the height of an object. The higher that an object is elevated, the greater the GPE. The relationship is expressed in the following manner:
PEgrav = mass x g x height
PEgrav = m x g x h
Where,
m is the mass of the object,
h is the height of the object
g is the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.