Question

Is the center of mass of the dumbbell in FIGURE Q12.1 at point a, b, or c? Explain.

Short answer

The center of mass is at the point$\mathbf{a}$.

Step-by-step solution

Step.1

Centre of mass is a point within the boundary of system at which the total mass of the system appears to be concentrated such that the system undergoes translator motion by the application of external force.

Step.2

The following figure shows the dumbbell.

Step.3

About the center of mass, the moment for $M$and $m$will be equal.

$\begin{array}{l}MR=mr\\ MR-mr=0\\ \frac{M}{m}=\frac{r}{R}\end{array}$

The ratios of the masses will be equal to the inverse ratio of the distances from the center of mass. From the figure, it is clear that $(M>m)$ and hence $(r>R)$. Therefore, the center of mass will be at point ' $\text{a}$' which satisfies the condition $(r>R)$.

Therefore, at the center of mass is at the point $\mathbf{a}$.