Question

If a mass $m$ is placed in the vicinity of another mass $M$, It experiences a gravitational force of attraction. However, if these two masses are at a very large distance, the force of attraction is negligible. The gravitation potential is the amount of work done per unit mass in bringing it slowly from infinity to some finite distance from $M$. Hence, gravitational potential is a state function rather than a path function.
One application of this potential is in finding the escape velocity of a body from earth. As we know the gravitation potential energy associated with mass $m$ on earth surface is $-\frac{GMm}{R}$. The mechanical energy conservation gives $\underset{–}{V_{escape}=\sqrt{\frac{2GM}{R}}}$ or $V_{escape}=\sqrt{2gR}$. An inquisitive mind decides to get the result by kinematics, considering a particle following curvilinear path from surface to infinity, gravity changes with height as $g^{\prime }=g{(1+\frac{h}{R})}^{-2}$ or $v\frac{dv}{dh}=-g{(1+\frac{h}{R})}^{-2}$. Solution of the equation within limits $h=0$ to $h=\infty$ gives $\underset{–}{V_{escape}=\sqrt{2gR}}$.
For most of the objects, either point masses or objects of finite dimension, the variation of potential in space exhibits symmetric behaviour. In case of spherical objects of uniform density, the locus of equipotential points is a spherical shell of any given radius. Hence, this potential field is symmetric about all three axes passing through the centre of spherical object. Consider a somewhat complicated objects as shown in the adjacent figure.
The figure shows a solid cube of edge length $10cm$. The origin is the centre of cube as shown. Eight spherical cavities are formed in this cube, each having a radius of $1cm$ and centers at $(\pm 2cm,\pm 2cm,\pm 2cm$). This figure shows wide range of equipotential surfaces / curves.
Which of the following can be inferred from the definition of gravitational potential energy in paragraph $1$?

• A. Gravitational potential energy is independent of total mass of system
• B. Gravitational potential is always a negative quantity
• C. Gravitational potential is negative if energy at infinity is assumed to be absolute zero
• D. None of the above

Answer 1

Answer: (C)