Question

Gravitational potential at the centre of a semi-circle formed from a thin wire of mass m and radius R is-
(A) $-2Gm/R$
(B) $-Gm/R$
(C) $-Gm/2R$
(D) $-Gm/3R$

Answer 1

Hint Gravitational potential energy is the energy stored by a unit test mass but if we know that the position of any entity changes because of the applied force then we can say that potential energy equals the amount of work done.

Complete step by step answer
In the above question following information is given-
Mass $=m$
Radius $=R$
Now, we need to calculate the gravitational potential. Firstly, we have to write the potential energy formula for a full circle-
Gravitational Potential at the centre of a full circle $=-2Gm/R$
The above equation is applicable only when we have to find the potential of a complete circle but here we have to find a semi-circle. For this we need to half the total gravitational energy and thereby we have the answer.
 $V=\dfrac{V}{2}$
Where, V is the symbol for gravitational potential.
$V'=\dfrac{V}{2}$
$Jk{{g}^{-1}}$
$-Gm/2R$
Here, V’ is used for the gravitational potential of a semicircle.

Therefore, the answer is clearly option (B).

Additional Information
Gravitational potential and electrostatic potential have a lot in common, both of them depend upon the separation that is R (radius). In both cases potential is defined as the work done in changing the separation between the interacting objects.
The main difference is the nature of the force –
In electrostatic force charges can be both negative or positive that is attractive or repulsive, but,
In potential the force of gravity is always attractive that is why it is written in negative terms.

Note
Gravitational potential is always negative since it is attractive in nature and considering unit mass do not confuse it with gravitational potential energy.
The SI unit is $Jk{{g}^{-1}}$ and is a scalar quantity.