Question

The centers of two identical spheres are $1.0\,m$ apart. If the gravitational force between the spheres is $1.0\,N$, then what is the mass of each sphere? ($G = 6.67 \times {10^{ - 11}}\,{m^3}k{g^{ - 1}}{s^{ - 2}}$).

Answer 1

Here we use the formula of the force of gravitational force which is defined as the force two bodies is directly proportional to the masses of the body and inversely proportional to the square of the distance between them and is given by
$F \propto \,\dfrac{{{m_1}{m_2}}}{{{r^2}}}$
$\Rightarrow \,F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}$
Here, $G$ is the gravitational force, $F$ is the gravitational force, ${m_1}$ and ${m_2}$ are the masses of the sphere and $r$ is the distance of the two spheres.

Complete step by step answer:
It is given the question that the distance between the masses of each sphere is $1.0\,m$.

Therefore, $r = \,1.0\,m$
Also, the force between the two spheres, $F = \,1.0\,N$
And, gravitational constant acting between the spheres, $G = \,6.67 \times {10^{ - 11}}\,{m^3}\,k{g^{ - 1}}{s^{ - 1}}$
Now, the force acting between the centers of two spheres can be calculated by using the formula of the gravitational force which is given by
$F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}$
Here, it is given in the question that the spheres are identical, therefore, the masses of the spheres should also be identical which means that ${m_1} = \,{m_2}\, = \,m$
Therefore, the above equation becomes
$F = G\dfrac{{m \times m}}{{{r^2}}}$
$F = G\dfrac{{{m^2}}}{{{r^2}}}$
Now, putting the values of $F$ , $G$ and $r$ , in the above equation, we get
$1.0\, = \,6.67 \times {10^{ - 11}} \times \,\dfrac{{{m^2}}}{{{{\left( 1 \right)}^2}}}$
$\Rightarrow \,{m^2} = \dfrac{{1.0}}{{6.67 \times {{10}^{ - 11}}}}$
$\Rightarrow \,{m^2} = \,0.1499 \times {10^{11}}$
$\Rightarrow \,{m^2} = 1.499 \times {10^{10}}$
Now, the value of the mass can be calculated by taking square root on both the sides
$\therefore m = \,1.224 \times {10^5}\,kg$

Hence, the mass of both the spheres is $1.224 \times {10^5}\,kg$.

Note: Here, $G = \,6.67 \times {10^{ - 11}}\,{m^3}\,k{g^{ - 1}}{s^{ - 1}}$ is the gravitational constant which is also known as Newton’s fundamental gravitational constant. In the above example, it tells us that the gravitational field acting between the two masses each of $1\,kg$ that are separated by a distance of one meter is $6.67 \times {10^{ - 11}}\,{m^3}\,k{g^{ - 1}}{s^{ - 1}}$ . This gravitational constant $G$ is used in both Newton’s and Einstein’s equations.