JEE Main 2026 — Rotational Motion Question with Solution

For two solid spheres in contact with the tangent line at the contact point.
Sphere 1: $m_1=5$ kg, $r_1=0.1$ m
Sphere 2: $m_2=10$ kg, $r_2=0.2$ m
For a solid sphere, $I_{cm}=\frac{2}{5}m{r}^2$
The tangent at contact is perpendicular to the line joining centers. The perpendicular distance from each sphere's center to this tangent line equals its radius.
Using parallel axis theorem: $I_{total}=I_{cm}+m{d}^2$
For sphere 1: $I_1=\frac{2}{5}(5)(0.1{)}^2+5(0.1{)}^2=0.02+0.05=0.07$ kg·m²
For sphere 2: $I_2=\frac{2}{5}(10)(0.2{)}^2+10(0.2{)}^2=0.16+0.4=0.56$ kg·m²
Total: $I=0.07+0.56=0.63$ kg·m²