Three identical spheres each of mass 2 M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4 ~m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is \frac4 \sqrt2x , where the value of x is \underline\hspace2cm .

Center of Mass PYQ — JEE Main 2024 · 1 Feb · Shift 1 · Q88 with Solution

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Correct answer: 3

Set up the coordinates

Since the triangle is right-angled with perpendicular sides of length $4\,\text{m}$ , and the point of intersection of these sides is taken as the origin, the three spheres are at:

(0,0),\quad (4,0),\quad (0,4)

Each sphere has equal mass $2M$ .

Use the centre of mass formula

For equal masses, the coordinates of the centre of mass are just the averages of the coordinates:

x_{\text{CM}} = \frac{0+4+0}{3} = \frac{4}{3}

y_{\text{CM}} = \frac{0+0+4}{3} = \frac{4}{3}

So the centre of mass is at

\left(\frac{4}{3},\frac{4}{3}\right)

Find the magnitude of the position vector

The magnitude of the position vector of the centre of mass from the origin is

r = \sqrt{\left(\frac{4}{3}\right)^2 + \left(\frac{4}{3}\right)^2}

r = \sqrt{\frac{16}{9} + \frac{16}{9}} = \sqrt{\frac{32}{9}} = \frac{4\sqrt{2}}{3}

Compare with the given form

Given,

r = \frac{4\sqrt{2}}{x}

Thus,

\frac{4\sqrt{2}}{x} = \frac{4\sqrt{2}}{3}

So,

x=3

Comparison with stored answer

Stored correct answer = $3$

Our derived answer also gives $x=3$ . Hence, they agree.

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