Question

Two spheres of mass $M$ and $4M$ are connected by a rod whose mass is negligible, and the distance between the centers of each sphere is $d$.
The $4M$ sphere is moved a distance $d/3$ towards the smaller sphere.
How far has the Center of Mass for the entire object moved?

• A. The Center of Mass has not moved, because both spheres still have their original masses
• B. $d/15$
• C. $d/5$
• D. $4d/15$
• E. $8d/15$

Answer 1

The correct option is D $4d/15$

Let mass $M$ is placed is at origin and mass $4M$ is on the x-axis at a distance of $d$ from origin ,

then centre of mass relative relative to $M$ is given by ,

$x_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$ ,

given $m_1=M,m_2=4M,x_1=0,x_2=d$ ,

$x_{cm}=\frac{M\times 0+4M\times d}{M+4M}=4d/5$ ,

when $x_{2}=d-d/3=2d/3$ ,

then $x_{cm}^{\prime }=\frac{M\times 0+4M\times \left(2d/3\right)}{M+4M}=8d/15$ ,

therefore movement in centre of mass ,

$=x_{cm}-x_{cm}^{\prime }=4d/5-8d/15=4d/15$