Two small and heavy spheres, each of mass $M$, are placed a distance $r$ apart on a horizontal surface. The gravitational potential at the mid-point on the line joining the centre of the spheres is
- Zero
- $ \frac{GM}{r} $
- $ -\frac{2GM}{r} $
- $ \frac{-4\,GM}{r} $
The Correct Option is D
Solution and Explanation
$A, V_{A}=\frac{-G M}{r / 2}$
Similarly, for the mass $M$ at $B$,
gravitation potential $V_{B}=-\frac{G M}{r / 2}$
Total potential $V_{A}+V_{B}=-\frac{G M}{r / 2}-\frac{G M}{r / 2}$
$=-\frac{4 G M}{r}$
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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.