Answer

Verified

53.7k+ views

**Hint**Write down the equation for acceleration due to gravity. Rearrange the equation and find the formula for mass. Then calculate the volume of earth. Finally substitute, mass and volume in the equation which states that the ratio of mass to volume gives us density.

**Complete step-by-step solution**

Acceleration due to gravity on earth is given as,

\[g = \dfrac{{GM}}{{{R^2}}}\,\,\, \ldots (1)\]

Where, g: Acceleration due to gravity

G: Gravitational constant

M: Mass of earth

R: Radius of earth

Now we know, density is given as,

\[density = \dfrac{{mass}}{{volume}}\]

\[\therefore p = \dfrac{M}{V}\,\,\, \ldots (2)\]

Volume of Earth is

\[V = \dfrac{4}{3}\pi {R^3}\,\, \ldots (3)\]

Now, rearranging equation.(1) gives us,

\[{\text{M = }}\dfrac{{{\text{g}}{{\text{R}}^{\text{2}}}}}{{\text{G}}}\,\,\,{\ldots (4)}\]

By substituting equation.(3) and equation.(4) in equation.(2) we get,

$p=\dfrac{{{\text{g}}{{\text{R}}^{\text{2}}}}}{ \dfrac{4}{3}\pi {R^3}}$

\[\therefore {\text{p = }}\dfrac{{{\text{3g}}}}{{{\text{4}\pi \text{GR}}}}\]

So, the relation between density and acceleration due to gravity is given as \[p = \dfrac{{3g}}{{4\pi GR}}\].

Acceleration due to gravity is proportional to mean density of earth and its radius.

**Hence, the correct answer is option (B).**

**Additional Information**

Equation of gravitational constant and acceleration due to gravity can be calculated as shown below:

Suppose, a body of mass m is placed on the surface of earth and assumes the shape of the earth to be round. If mass of the earth is M and radius of earth is R, then by Newton’s law of gravitation we get,

\[{\text{F = G}}\dfrac{{{\text{Mm}}}}{{{{\text{R}}^{\text{2}}}}}\,\,{ \ldots (1)}\]

Now, from Newton’s Second law of motion,

\[{\text{F = mg}}\,\,{ \ldots (2)}\]

From equation.(1) and equation.(2) we get,

\[{\text{mg = G}}\dfrac{{{\text{Mm}}}}{{{{\text{R}}^{\text{2}}}}}\]

\[\therefore {\text{g = }}\dfrac{{{\text{GM}}}}{{{{\text{R}}^{\text{2}}}}}\,\,\]

**Note**This expression can also be used at and within the earth at distance R from the centre of the earth. Generally, we use P (rho) as the symbol for density but in this question it is denoted by symbol p. So don’t get confused by the initialization. Solve the question using the initials mentioned in the question.

Recently Updated Pages

Why do we see inverted images in a spoon class 10 physics JEE_Main

A farsighted man who has lost his spectacles reads class 10 physics JEE_Main

State whether true or false The outermost layer of class 10 physics JEE_Main

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

State two factors upon which the heat absorbed by a class 10 physics JEE_Main

What will be the halflife of a first order reaction class 12 chemistry JEE_Main

Other Pages

The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

A given ray of light suffers minimum deviation in an class 12 physics JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Oxidation of succinate ion produces ethylene and carbon class 12 chemistry JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main