As, Is=(2)/(5) M Rs^2, Ih=(2)/(3) M Rh^2 As Is=Ih ∴ (2)/(5) M Rs^2=(2)/(3) M Rh^2 ∴(Rs)/(Rh)=

step 1 As per the question, two spheres are there. One is solid sphere and the other one is holo sphere

As, Is=(2)/(5) M Rs^2, Ih=(2)/(3) M Rh^2 As Is=Ih ∴ ...

$\mathrm{As}, I_{s}=\frac{2}{5} M R_{s}^{2}, I_{h}=\frac{2}{3} M R_{h}^{2}$
As $\quad I_{s}=I_{h}$
$\therefore \quad \frac{2}{5} M R_{s}^{2}=\frac{2}{3} M R_{h}^{2}$
$\therefore$
$\frac{R_{s}}{R_{h}}=\frac{\sqrt{5}}{\sqrt{3}}$

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Key Concepts

Algebraic Manipulation in Physics

Algebraic manipulation is an important technique in physics, used to rearrange equations and isolate desired variables. In the context of equating moments of inertia, it involves canceling common factors and taking square roots to derive meaningful relationships between different physical quantities.

Moment of Inertia for Rigid Bodies

Different geometric shapes have characteristic formulas for their moment of inertia, which reflect how their mass is spread relative to the axis of rotation. Standard formulas, such as those for a solid sphere and a solid hemisphere, provide essential tools for analyzing and comparing the rotational characteristics of various objects.

Rotational Dynamics

Rotational dynamics is the branch of mechanics that deals with the motion of bodies rotating about an axis. It involves understanding how torque, angular momentum, and moment of inertia interact to govern the rotational behavior of objects.

Moment of Inertia

Moment of inertia is a fundamental concept in rotational dynamics that quantifies an object's resistance to angular acceleration about a specific axis. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation, playing a role similar to mass in linear motion.

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