Question

The ratio of the radius of a uniform circular disc and its radius of gyration about a tangent in its plane is

• A. \(\sqrt{5} : 2\)
• B. \(2 : \sqrt{5}\)
• C. \(4 : 5\)
• D. \(5 : 4\)

Hint:

Use the parallel axis theorem when the axis is not through the center of mass. For a disc: \[ I_{\text{tangent}} = I_{\text{CM}} + MR^2 \quad \text{and} \quad k = \sqrt{\frac{I}{M}} \]

Answer 1

The Correct Option is B

Step 1: Moment of inertia of a disc about a tangent in its plane.
Using the parallel axis theorem, moment of inertia \(I = I_\text{CM} + MR^2\) \[ I_\text{tangent} = \frac{1}{4}MR^2 + MR^2 = \frac{5}{4}MR^2 \] Step 2: Radius of gyration \(k\) is defined by: \[ I = Mk^2 \Rightarrow k = \sqrt{\frac{5}{4}}R = \frac{\sqrt{5}}{2}R \] Step 3: Ratio of radius to radius of gyration: \[ \frac{R}{k} = \frac{R}{\frac{\sqrt{5}}{2}R} = \frac{2}{\sqrt{5}} \] So, the required ratio is \( \boxed{2 : \sqrt{5}} \)