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Question: The gravitational constant G is equal to \(6.67 \times {10^{ - 11}}N{m^2}k{g^{ - 2}}\) in vacuum. It...

The gravitational constant G is equal to 6.67×10−11Nm2kg−26.67 \times {10^{ - 11}}N{m^2}k{g^{ - 2}} in vacuum. Its value in a dense matter of density 1020gcm3{10^{20}}\dfrac{g}{{c{m^3}}} will be
A) 6.67×10−1Nm2kg−2{10^{ - 1}}N{m^2}k{g^{ - 2}}
B) 6.67×10−11Nm2kg−2{10^{ - 11}}N{m^2}k{g^{ - 2}}
C) 6.67×10−21Nm2kg−2{10^{ - 21}}N{m^2}k{g^{ - 2}}
D) None of these

Explanation

Solution

Newton has given a gravitational law and it states that the force of gravitation is directly proportional to the product of the mass of object and inversely to the square of the distance between them Fgrav∝m1×m2r2{F_{grav}}\propto \dfrac{{{m_1} \times {m_2}}}{{{r^2}}}. Hence the gravitational constant remains constant whether it is in denser or rarer medium.

Step by step solution:
Step 1:
Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center.
All objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers.

Step 2:
Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as: Fgrav∝m1×m2r2{F_{grav}}\propto \dfrac{{{m_1} \times {m_2}}}{{{r^2}}} where, Fgrav{F_{grav}}is the force, m1{m_1}is the mass of one body, m2{m_2}is the mass of second body, r2{r^2}is the distance between then and squared.
Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality as F=G(m1)(m2)r2F = \dfrac{{G\left( {{m_1}} \right)\left( {{m_2}} \right)}}{{{r^2}}} here G is the proportionality constant and value of G is 6.67×10−11Nm2kg−26.67 \times {10^{ - 11}}N{m^2}k{g^{ - 2}}
Now coming to the question:
We have been asked what will be the value of Gravity G when a body is placed into a dense medium.
So here is a diagram of the given question:


So from the above formula of law of gravitation F=G(m1)(m2)r2F = \dfrac{{G\left( {{m_1}} \right)\left( {{m_2}} \right)}}{{{r^2}}} we can see that the gravity does not depend on the medium and hence the value of G will not change.
The value of G will remain to 6.67×10−11Nm2kg−26.67 \times {10^{ - 11}}N{m^2}k{g^{ - 2}}

So, option B is the correct answer.

Note: In the above question we are asked that two bodies is in a dense medium but in simple words we can answer this question or any other such type of question that force of gravitation is only dependent on the distance of the body and it is inversely proportional to it and directly related to mass of bodies. It does not change with change in place, medium, or body etc.