Question

The value of universal gravitational constant G is-
A. \[6.67\times {{10}^{-11}}\dfrac{N{{m}^{2}}}{kg}\]
B. \[6.67\times {{10}^{-11}}\dfrac{N{{m}^{2}}}{k{{g}^{2}}}\]
C. \[66.7\times {{10}^{-11}}\dfrac{N{{m}^{2}}}{k{{g}^{2}}}\]
D. \[66.7\times {{10}^{-11}}\dfrac{N{{m}^{2}}}{kg}\]

Answer 1

Hint: We can find the value of the gravitational constant, the universal law of gravitation. If we are not applying any other force on an object, its weight will be the force. The parameters of the earth can be used to find the gravitational constant.

Formula used: \[F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\], where G is the universal gravitational constant, \[{{m}_{1}}\]and \[{{m}_{2}}\] are the masses of the objects and \[r\] is the distance between them.

Complete step-by-step answer:
According to the universal law of gravitation, everybody in the universe attracts each other and this force of attraction is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This definition can be expressed as,
\[F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\], where G is the universal gravitational constant, \[{{m}_{1}}\]and \[{{m}_{2}}\] are the masses of the objects and \[r\] is the distance between them.
To find the value of gravitational constant, we can rewrite this equation as,
\[G=\dfrac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}\]
If we are considering \[{{m}_{1}}\] as the mass of the earth (M) and \[{{m}_{2}}\] as the mass of the object (m), then the force will be equal to the weight of that object. That is mg. Since no other forces are acting on the object. We can assign these details to the equation.
\[G=\dfrac{mg{{r}^{2}}}{mM}\]
\[G=\dfrac{g{{r}^{2}}}{M}\]
We are considering the object is placed on the surface of the earth. So we can take the distance as the radius of the earth.
The radius of the earth, \[r=6371km=6.37\times {{10}^{6}}m\]
Mass of the earth, \[M=5.972\times {{10}^{24}}kg\]
Acceleration due to gravity, \[g=9.8m{{s}^{-2}}\]
We can plug these values into the equation.
\[G=\dfrac{9.8\times {{(6.37\times {{10}^{6}})}^{2}}}{5.972\times {{10}^{24}}}\]
\[G=6.66\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}}\]
Slight differences will occur. It is purely depending upon the mass and radius of the earth. It is the same in all the universe. That’s why it is treated as a universal gravitational constant. Its actual value is \[6.67\times {{10}^{-11}}\dfrac{N{{m}^{2}}}{k{{g}^{2}}}\]
So the correct option is B.

Additional information:
Here we have mentioned the gravitational force. Now we can discuss some properties of gravitational force. It is the weakest force in our nature. As we know it is always attractive. Whereas electrostatic and magnetic forces can be attractive or repulsive according to the nature of the body. It is considered as a mutual force. That is both bodies involved in this force are attracted with the same force. Gravitational force is a central force and it is depending on the mass and distance between the bodies. It does not depend upon the medium between the interacting bodies, hence there is no gravitational shielding available. This force range is very long.

Note: The value of G also does not depend upon the medium between the two bodies. Universal gravitational constant G and acceleration due to gravity g are different. Si units of these two quantities are also different. SI unit of gravitational constant is \[N{{m}^{2}}k{{g}^{-2}}\], whereas the SI unit of acceleration due to gravity is \[m{{s}^{-2}}\].