Question

If gravitational potential at a place is $V = (4xy - 2yz + 4xyz)$ J/Kg then find gravitational field intensity at potential (1m, 1m, 1m);
A) $9$ N/Kg
B) $\sqrt {104}$ N/Kg
C) $\sqrt {102}$ N/Kg
D) $10$ N/Kg

Answer 1

Above question is based on the relation of the Gravitational Potential and Gravitational field intensity.
Gravitational Potential is amount of work done in bringing unit mass from an infinite place, which is given by the formula;
$V = \dfrac{{ - GM}}{r}$ (V is the gravitational potential, G is the gravitational constant, M is the mass of the bo\partial y)
Gravitational field Intensity is the strength of a gravitational field which is applied on a unit mass, which is given by the formula,
$I = \dfrac{{GM}}{{{r^2}}}$ (I is the intensity, G is the gravitational constant, M is the mass, r is the distance between the infinity and the place of work)
Gravitational field intensity and Gravitational potential has the relation:
$I = - \dfrac{{\partial \mathop V\limits^ \to }}{{\partial r}}$
Using the above relations we will solve the given problem.

Complete step by step solution:
Let us explain Gravitational field intensity and Gravitational potential in more detail first.
Gravitational Potential: Gravitational potential at a point in the gravitational field is the amount of work done in bringing a unit mass from infinity to that point. Gravitational potential difference is defined as the work done to move a unit mass from one point to the other point in the gravitational field.
Gravitational field intensity: Gravitational field intensity is the strength of a Gravitational field which is applied on a unit test mass. A gravitational force's intensity depends upon the source mass and the distance of unit test mass from the source mass.
Now, we will do the calculation part of the problem;
Gravitational potential is given in the question as;
$V =4xy -2yz + 4xyz$
Gravitational field intensity is then given by;
$\Rightarrow I = - [\dfrac{{\partial V}}{{\partial x}}\mathop i\limits^ \wedge + \dfrac{{dV}}{{\partial y}}\mathop j\limits^ \wedge + \dfrac{{dV}}{{\partial z}}\mathop k\limits^ \wedge ]$ (as the gravitational potential is three dimensional, so we have differentiated into three components)
$\Rightarrow I = - [\dfrac{{\partial (4xy - 2yz + 4xyz)}}{{\partial x}}\mathop i\limits^ \wedge + \dfrac{{\partial (4xy - 2yz + 4xyz)}}{{\partial x}}\mathop j\limits^ \wedge + \dfrac{{\partial (4xy - 2yz + 4xyz)}}{{\partial x}}\mathop {k]}\limits^ \wedge$
$\Rightarrow I = - [(4y + 4yz)\mathop i\limits^ \wedge + (4x - 2z + 4xz)\mathop j\limits^ \wedge + ( - 2y + 4xy)\mathop {k]}\limits^ \wedge$(differentiated the entire term with x, y and z)
We have to find Intensity at the points (1, 1, 1)
$\Rightarrow I = - [(4 + 4)\mathop i\limits^ \wedge + (4 - 2 + 4)\mathop j\limits^ \wedge + ( - 2 + 4)\mathop {k]}\limits^ \wedge$(We substituted x, y, z as 1)
$\Rightarrow I = - [(8)\mathop i\limits^ \wedge + (6)\mathop j\limits^ \wedge + (2)\mathop {k]}\limits^ \wedge$
Magnitude of the Intensity is;
$\Rightarrow I = \sqrt {{8^2} + {6^2} + {2^2}}$
$\Rightarrow I = \sqrt {104}$ N/Kg

Hence, Option B is correct.

Note: The phenomenon of gravitational field intensity is clearly seen by us in our solar system, galaxies, moon, earth and other planets. Even we and every other creature are attracted towards earth due to the gravitational force or pull otherwise we the human beings and every other creature would have floated in the air if there was no gravitational force between us and earth.