Two identical solid spheres each of mass 2 kg and radii 10 cm are… — Solution | JEE Physics

NVAEasyJEE 2023Theorems of M.I (Parallel & Perpendicular Axis)

Two identical solid spheres each of mass $2 \, \text{kg}$ and radii $10 \, \text{cm}$ are fixed at the ends of a light rod. The separation between the centers of the spheres is $40 \, \text{cm}$ . The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is $\times 10^{-3} \, \text{kg}\cdot\text{m}^2$ :

Answer

Correct answer:176

Step-by-step solution

Using parallel axis theorem

Given: Each solid sphere has mass $m = 2 \, \text{kg}$ and radius $r = 10 \, \text{cm} = 0.1 \, \text{m}$ . The separation between centers is $40 \, \text{cm}$ , so the distance of each sphere from the midpoint is $d = 20 \, \text{cm} = 0.2 \, \text{m}$ .

Find: The moment of inertia of the system about an axis perpendicular to the rod through its midpoint.

Using parallel axis theorem,

I_{\text{sys}} = \left(\frac{2}{5}mr^2 + md^2\right) \times 2

Substituting the values,

I_{\text{sys}} = \left(\frac{2}{5} \times 2 \times 0.01 + 2 \times 0.04\right) \times 2

= (0.008 + 0.08) \times 2 = 0.088 \times 2 = 0.176 \, \text{kg}\cdot\text{m}^2

Therefore,

I_{\text{sys}} = 176 \times 10^{-3} \, \text{kg}\cdot\text{m}^2

So the required numerical value is 176.

Common mistakes

Using $d = 0.4 \, \text{m}$ instead of $d = 0.2 \, \text{m}$ . The parallel axis theorem requires the distance from each sphere's center to the axis through the midpoint, not the full separation between the two spheres.
Ignoring the sphere's own moment of inertia $\frac{2}{5}mr^2$ . Each sphere is not a point mass, so both the self-rotation term and the shift term $md^2$ must be included.
Calculating the moment of inertia for only one sphere and forgetting to multiply by $2$ . The system contains two identical spheres placed symmetrically about the axis.

Practice more Theorems of M.I (Parallel & Perpendicular Axis) questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Theorems of M.I (Parallel & Perpendicular Axis)

Answer

Step-by-step solution

Common mistakes

Practice more Theorems of M.I (Parallel & Perpendicular Axis) questions

Related questions