A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii and masses of both the bodies are identical and the ratio of their kinetic energies is $$\frac{7}{x}$$, where x is _____.

For a body rolling down an incline from rest, the total kinetic energy ($$KE$$) at the bottom is equal to the initial gravitational potential energy.

Energy Conservation: Since both bodies descend the same vertical height $$h$$:

$$KE_{\text{ring}} = M_{\text{ring}}gh$$

$$KE_{\text{sphere}} = M_{\text{sphere}}gh$$

Assuming identical masses for comparison:

$$KE_{\text{ring}} = KE_{\text{sphere}}$$

$$\frac{KE_{\text{ring}}}{KE_{\text{sphere}}} = 1$$

Given the ratio is $$\frac{7}{x}$$:

$$\frac{7}{x} = 1 \implies x = 7$$