Question 309: Centre of Mass and Rotational Motion

Question

Three identical metal balls each of radius ' $r$ ' are placed such that an equilateral triangle is formed when centres of three ball are joined. The centre of mass of the system is located at

Accepted Answer

Three identical metal balls, each with radius $r$, are arranged such that the centers of these balls form an equilateral triangle. The center of mass of this system can be determined as follows:
Since the balls are identical with equal volumes and densities, their masses are also the same. This simplifies the calculation of the center of mass.
The expressions for the coordinates of the center of mass, $\mathrm{X}_{\mathrm{cm}}$ and $\mathrm{Y}_{\mathrm{cm}}$, are derived as follows:
$ \begin{aligned} & \mathrm{X}_{\mathrm{cm}} = \frac{mx_1 + mx_2 + mx_3}{m + m + m} = \frac{x_1 + x_2 + x_3}{3}, \ & \mathrm{Y}_{\mathrm{cm}} = \frac{my_1 + my_2 + my_3}{m + m + m} = \frac{y_1 + y_2 + y_3}{3}. \end{aligned} $
Here, $\mathrm{X}_{\mathrm{cm}}$ and $\mathrm{Y}_{\mathrm{cm}}$ represent the centroid of the equilateral triangle formed by the centers of the three balls. Since all masses are equal and symmetrically distributed around the centroid, the center of mass coincides with the centroid of the triangle, which is also the point where the medians intersect.

Answer

centre of one of the balls.

Answer

point of intersection of medians.

Answer

line joining centres of any two balls.

Answer

on the circumference of any one of the balls.

Centre of Mass and Rotational Motion

Three identical metal balls each of radius ' r ' are placed such that an equilateral triangle is formed when centres of three ball are joined. The centre of mass of the system is located at

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