Question

Consider two solid uniform spherical objects of the same density $\rho$. One has radius R and the other has radius $2$R. They are in outer space where the gravitational fields from other objects are negligible. If they are arrange with their surface touching, what is the contact force between the objects due to their traditional attraction?

• A. $G{\pi }^2{R}^4$
• B. $\frac{128}{81}G{\pi }^2{R}^4{\rho }^2$
• C. $\frac{128}{81}G{\pi }^2$
• D. $\frac{128}{87}{\pi }^2{R}^{2}G$

Answer 1

Answer: (B)

Given Data

Radius of sphere 1 is R

Radius of sphere 2 is 2R

Density in P

Distance between Two sphere in $r\Rightarrow R+2R$

$=3R...\left(1\right)$

We know that

Density $=\frac{mass}{volume}$ [For uni form spherical objects]

$\therefore$ mass = volume $\times$ Density

volume of sphere in given be $=\frac{4}{3}\pi R^3$

$\therefore$ mass of sphere 1 in $m_1\Rightarrow \frac{4}{3}\pi R^3\rho ...\left(2\right)$

mass of sphere 2 in $m_2\Rightarrow \frac{4}{3}\pi (2R{)}^3\rho ...(3)$

$\to$ From Theorem of Gravitation

$F=G\frac{m_1{m}_2}{r^2}\Rightarrow G\frac{\left(\frac{4}{3}\pi R^3\rho \right)\left(\frac{4}{3}\pi \right(8R^3\left)\rho \right)}{(3R{)}^2}$

$F\Rightarrow \frac{128}{81}G{\pi }^2{R}^4{\rho }^2$

Hence $AnswerisB$