Question

Radius of gyration of a disc about its diameter is?

• A. \( \frac{r}{\sqrt{2}} \)
• B. \( \frac{r}{\sqrt{3}} \)
• C. \( \frac{r}{2} \)
• D. \( \frac{r}{4} \)

Hint:

The radius of gyration for common shapes like discs can be easily found by using the formula for the moment of inertia and applying it to the definition of radius of gyration.

Answer 1

The Correct Option is B

The radius of gyration \( k \) of a body about an axis is defined as the distance from the axis at which the body's mass can be considered to be concentrated without changing its moment of inertia. For a disc rotating about its diameter, the formula for the radius of gyration is given by: \[ k = \sqrt{\frac{I}{m}} \] where: - \( I \) is the moment of inertia of the disc about the axis (diameter), - \( m \) is the mass of the disc. For a disc, the moment of inertia about the diameter is: \[ I = \frac{1}{4} m r^2 \] Substituting into the equation for \( k \): \[ k = \sqrt{\frac{\frac{1}{4} m r^2}{m}} = \frac{r}{\sqrt{3}} \] Thus, the radius of gyration is \( \frac{r}{\sqrt{3}} \).