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Standard XII

Physics

Center of Mass as an Average Point

A dumbbell is...

Question

A dumbbell is composed of two spheres that are connected by a rod whose mass is negligible. The distance between the centers of each sphere is $d$ . One of the spheres has a mass of $2 M$ and the mass of the other sphere is unknown, as shown.
If the dumbbell can balance when supported at a point $2 d / 3$ from the center of the $2 M$ sphere, what is the mass of the second sphere?

2 M

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3 M

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2 \sqrt{2} M

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4 M

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5 M

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Solution

The correct option is A $4 M$
Point C represents the centre of mass of the system i.e $x = \frac{2 d}{3}$
Given : $m_{1} = 2 M$
Using $x = \frac{m_{1} x_{1} + m_{2} x_{2}}{m_{1} + m_{2}}$

$∴$ $\frac{2 d}{3} = \frac{(2 M) \times 0 + m_{2} d}{2 M + m_{2}}$

OR $4 d M + 2 d m_{2} = 3 d m_{2}$ $⟹ m_{2} = 4 M$

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The correct option is A 4MPoint C represents the centre of mass of the system i.e x=2d3Given : m1=2M Using x=m1x1+m2x2m1+m2∴ 2d3=(2M)×0+m2d2M+m2 OR 4dM+2dm2=3dm2 ⟹m2=4M