Question

Two solid spheres of radii \( r_1 \) and \( r_2 \) (\( r_2>r_1 \)) made of the same material are kept in contact. The distance of their center of mass from their point of contact is:

• A. \( \frac{r_1^3 (r_1 + r_2)}{r_1^3 + r_2^3} \)
• B. \( \frac{r_2 (r_1 + r_2)}{r_1^2 + r_2^2} \)
• C. \( \frac{r_1^4 + r_2^4}{r_1^3 + r_2^3} \)
• D. \( \frac{r_2^3 (r_1 + r_2)}{r_1^3 + r_2^3} \)

Hint:

For objects of uniform density, their masses are proportional to their volumes. The center of mass is calculated using mass-weighted distances.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept of Center of Mass

- The spheres are made of the same material, meaning their masses are proportional to their volumes. - The mass of a sphere is given by: \[ m = \rho \times \frac{4}{3} \pi r^3 \] where \( \rho \) is the density of the material.
Step 2: Center of Mass Formula for Two Particles

- The center of mass for two objects is given by: \[ X_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
Step 3: Substituting Masses of Spheres

Since mass is proportional to \( r^3 \): \[ X_{\text{cm}} = \frac{r_2^3 \times (r_1 + r_2)}{r_1^3 + r_2^3} \]
Step 4: Conclusion

Since the distance of the center of mass from the point of contact is \( \frac{r_2^3 (r_1 + r_2)}{r_1^3 + r_2^3} \), Option (4) is correct.