Question

Radius of gyration of a solid cylinder of radius R and length L about its long axis of symmetry is

• A. R
• B. 2R
• C. {\sqrt2}R
• D. $\frac -R{2} $
• E. $\frac R{\sqrt2} $

Answer 1

Answer: (E)

The radius of gyration \( k \) of a body is defined as the distance from the axis of rotation at which the body's mass can be considered to be concentrated without affecting its rotational inertia. It is related to the moment of inertia \( I \) and the mass \( M \) by the formula:

\[ I = M k^2 \] For a solid cylinder rotating about its long axis of symmetry, the moment of inertia is: \[ I = \frac{1}{2} M R^2 \] Where: - \( M \) is the mass of the cylinder, - \( R \) is the radius of the cylinder. Using the formula for the radius of gyration: \[ k^2 = \frac{I}{M} \] Substituting the expression for \( I \): \[ k^2 = \frac{\frac{1}{2} M R^2}{M} = \frac{1}{2} R^2 \] Taking the square root: \[ k = \frac{R}{\sqrt{2}} \] Therefore, the radius of gyration of the solid cylinder about its long axis of symmetry is: \[ k = \frac{R}{\sqrt{2}} \]

Correct Answer:

Correct Answer: (E) \( \frac{R}{\sqrt{2}} \)