Question

For the three dumbbells shown above, the rod connecting them has a negligible mass and has a length of $d$. The spheres shown in Dumbbell 1 and Dumbbell 2 have a uniform density of $\rho$ and the spheres in Dumbbell 3 have a density of $2\rho$.
For which dumbbell is the center of mass furthest to the left?

• A. Dumbbell 1
• B. Dumbbell 2
• C. Dumbbell 3
• D. The location of the center of mass is the same for all three dumbbells

Answer 1

Answer: (D)

The location of the center of mass is the same for all three dumbbells

The centre of mass of the two body system (when both masses are equal) is given by ,

$x_{cm}=x_1+x_2/2$ ,

where $x_1,x_2$ are the distances of masses from origin respectively , let the first mass is at origin then ,

$x_1=0$ and $x_2=d$

where $d=$ distance between two masses ,

for given diagrams , the masses are equal in all three diagrams therefore centre of mass in each case will be ,

$x_{cm}=0+d/2=d/2$ ,

the location of CM is same for all three dumbells .