Question

In a certain unit, the radius of gyration

of a uniform disc about its central and

transverse axis is 2 5. . Its radius of

gyration about a tangent in its plane (in

the same unit) must be
(A) √5 (B) 2.5

(C) 2√2.5(D) √12.5​

Answer 1

Given:

The radius of gyration of a uniform disc about its central and transverse axis =2.5 units

To Find :

The radius of gyration of the uniform disc about a tangent in its plane = ?

Solution :

Since we know that the moment of inertia disc about central transverse axis is given as :

$= \frac{mR^2}{2}$

Here m and R are mass and radius of disc respectively .

Let ,  K = radius of gyration

∴ $mK^2 = \frac{mR^2}{2}$

So, $K = \frac{R}{\sqrt2}$  = 2.5 units    (radius of gyration is given 2.5 units )

Now by theorem of perpendicular axis :

$I_x + I_y = I_z$

Where $I_x , I_y$ and $I_z$ are the moment of inertia about x , y and z axis .

∴For a uniform disc , $I_x = I_y$

So, $I_x = I_y = \frac{I_z}{2}$ $= \frac{mR^2}{2\times 2} = \frac{mR^2}{4}$

Now by theorem of parallel axis :

Moment of inertia about tangent = $\frac{mR^2}{4} + mR^2 =\frac{5mR^2}{4}$

Let K' be the radius of gyration about the tangent , then :

$mK'^2 = \frac{5mR^2}{4}$

So, K' =$\frac{\sqrt5 R}{2}$

Since $\frac{R}{\sqrt2} = 2.5$ units

So, $K' = \frac{\sqrt5}{\sqrt2} \times 2.5$

           =$\sqrt{12.5}$ units

Hence , (D) is the correct option i.e. radius of gyration about a tangent of the disc is $\sqrt{12.5}$ units .

Answer 2

Explanation:

the answer is option b 2.5