Question

Two cylinders, one of which is hollow and other solid, have same mass and same moment of inertia about their respective geometrical axis. The ratio of their radii is?​

Answer 1

Answer : The ratio of their radii is √2 / 1 Step-by-step Explanation : Given : Two cylinders, one of which is hollow and other solid, have same mass and same moment of inertia about their respective geometrical axis. To find : Ratio of their radii = ? We know that , Moment of Inertia of a solid Cylinder about geometrical axis is given by , Is = 1/2 × M(Rs)² -------------(1) Where Rs = Radius of solid Cylinder Also, Moment of Inertia of a Hollow Cylinder about its geometrical axis is given by, Ih = M(Rh)² ----------------(2) Where , Rh = Radius of Hollow Cylinder According to given conditions, Is = Ih 1/2 × M(Rs)² = M(Rh)² (Rs)² = 2 × (Rh)² (Rs)²/ (Rh)² = 2 / 1 By taking square root on both sides we get, Rs / Rh = √2 / 1 Hence the ratio of their radii is √2 / 1

Answer 2

Step-by-step Explanation:

Given: Hollow and solid cylinders with same mass and moment of inertia

To Find: Ratio of the radii of the two cylinders

Solution:

• Finding the ratio of the two radii

The moment of inertia of the hollow cylinder $I_{h}$ is given by

$I_{h} = MR_{h}^{2} \cdots \cdots \cdots (1)$

where $M$ is the mass and $R_{h}$ is the radius of the cylinder

While for a solid cylinder, $I_{s} = MR_{s}^{2} \cdots \cdots \cdots (2)$ ($R_{s}$ is the radius).

Since it is given that the moment of inertia of both cylinders are equal, therefore,  $I_{h} = I_{s}$

$\Rightarrow MR_{h}^{2} = \frac{1}{2} MR_{s}^{2}$

$\Rightarrow \frac{R_{h}^{2}}{R_{s}^{2}} = \frac{1}{2}$

$\Rightarrow \frac{R_{h}}{R_{s}} = \frac{1}{\sqrt{2}}$  which is the required ratio

Hence, the ratio of the radii of the two cylinders is $\frac{R_{h}}{R_{s}} = \frac{1}{\sqrt{2}}$