Question

The gravitational potential energy is maximum at
$A.$ Infinity
$B.$ the earth’s surface
$C.$ the centre of the Earth
$D.$ twice the radius of the Earth

Answer 1

Hint Here we will proceed by explaining the meaning of gravitational potential energy, then by using the derivation of the gravitational potential energy equation we will get our answer.

Complete step-by-step answer:
Gravitational Potential energy – It is the potential energy held by an object because of its high position compared to a lower position. We can also say that the energy associated with gravitational force or gravity.The example of the gravitational potential energy is that if a pen being held above a table will have a higher gravitational potential then a pen sitting on the table.
When a body with mass m is moved from infinity to a point inside the gravitational influence of a source with mass M without accelerating it, the amount of work done in displacing it into the source field is stored in the form of potential energy this is known as gravitational energy. It is represented with the symbol of $Ug$.
Gravitational potential energy formula –
The equation of gravitational energy is:
$\Rightarrow G.P.E. = m \times g \times h$
Where, m is the mass in kilograms,
$g$ = acceleration due to gravity (9.8 on earth)
$h$ = the height above the ground in meters.
Gravitational potential energy is measured in joules J.
Derivation of Gravitational Potential Energy equation
Considering a source mass $'M'$ is placed at a point along the x-axis, initially, a test mass ‘m’ is at infinity. Let us consider a small amount of work is done in bringing source mass without acceleration through a very small distance (dx) which is given by -
$dw = Fdx$
Here, F is an attractive force and the displacement id towards the negative x-axis direction so F and dx are in the same direction. Then,
$dw = \dfrac{{\left( {GMm} \right)}}{{{x^2}}}dx$
Integrating both sides we get,
$w = \int\limits_\infty ^r {\dfrac{{GMm}}{{{x^2}}}} dx \\\ w = - {\left[ {\dfrac{{GMm}}{x}} \right]_\infty }^r \\\ w = - \left[ {\dfrac{{GMm}}{r}} \right] - \left( {\dfrac{{ - GMm}}{\infty }} \right) \\\ w = \dfrac{{ - GMm}}{r} \\\$
Since this work done is stored as its potential energy U, therefore gravitational potential energy U, therefore gravitational potential energy at a point which is at a distance $'r'$from the source mass is given by;
$U = \dfrac{{ - GMm}}{r}$
The negative sign indicates that the formula as the distance between two bodies increases the potential energy will tend to become less negative, that is it increases and it will become maximum equal to zero at infinity. So maximum potential energy will be when the two objects are placed at infinite distance from each other.
Hence, its maximum value is zero at infinity.

Note: Whenever we come up with this type of question related to potential energy one must know that the higher up an object is the greater its gravitational potential energy. The larger the distance something falls through the greater the amount of gravitational potential energy the object loses as it falls. Then by explaining the derivation of the gravitational potential equation we will get our answer.