The angular speed of a body changes from ω1 to ω2 without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :

Question

The angular speed of a body changes from ω1 to ω2 without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is : (A) ω2-ω1 (B) ω1: ω2 (C) √ω1: √ω2 (D) √ω2: √ω1

Tardigrade Admin · Accepted Answer

In the absence of external torque angular momentum is conserved.
The radius of gyration

k

of a body about a given line is defined as

I = M k^{2}

where,

I

is its moment of inertia and

M

is its total mass.
Also from law of conservation of angular momentum

J = I ω =

constant

(M k_{1}^{2}) ω_{1} = (M k_{2}^{2}) ω_{2}

\Rightarrow \frac{k _{1}^{2}}{k _{2}^{2}} = \frac{ω _{2}}{ω _{1}}

\Rightarrow \frac{k _{1}}{k _{2}} = \frac{ω _{2}}{ω _{1}}

Q. The angular speed of a body changes from $ω_{1}$ to $ω_{2}$ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :

$ω_{2} - ω_{1}$

$ω_{1} : ω_{2}$

$ω_{1} : ω_{2}$

$ω_{2} : ω_{1}$

Q. The angular speed of a body changes from ω1​ to ω2​ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :

ω2​−ω1​

ω1​:ω2​

ω1​​:ω2​​

ω2​​:ω1​​

Q. The angular speed of a body changes from $ω_{1}$ to $ω_{2}$ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :

$ω_{2} - ω_{1}$

$ω_{1} : ω_{2}$

$ω_{1} : ω_{2}$

$ω_{2} : ω_{1}$