The radius of gyration of an uniform rod of length l about an axis passing through one of its ends and perpendicular to its length is
Understand the Problem
The question is asking for the radius of gyration of a uniform rod of specified length when analyzed about a specific axis. This involves principles from mechanics and requires understanding the concept of radius of gyration related to mass distribution along the length of the rod.
Answer
The radius of gyration is $k = \frac{l}{\sqrt{3}}$.
Answer for screen readers
The radius of gyration of the uniform rod about the given axis is given by $k = \frac{l}{\sqrt{3}}$.
Steps to Solve
- Understanding Radius of Gyration
The radius of gyration ($k$) is defined as the square root of the ratio of the moment of inertia ($I$) to the total mass ($m$) of the object. [ k = \sqrt{\frac{I}{m}} ]
- Calculate the Moment of Inertia
For a uniform rod of length (l) about an axis passing through one of its ends and perpendicular to its length, the moment of inertia is given by: [ I = \frac{1}{3} ml^2 ]
- Identify the Mass of the Rod
For a uniform rod, the mass can be represented as: [ m = \rho \cdot V ] where $\rho$ is the density and (V) is the volume. For a rod of cross-sectional area (A) and length (l): [ V = Al ] So, (m) can be expressed in terms of (l) as (m = \rho A l).
- Combine the Equations
Now substituting (I) and (m) into the radius of gyration formula: [ k = \sqrt{\frac{\frac{1}{3} ml^2}{m}} = \sqrt{\frac{1}{3} l^2} ]
- Simplify the Radius of Gyration
This simplifies to: [ k = \frac{l}{\sqrt{3}} ]
The radius of gyration of the uniform rod about the given axis is given by $k = \frac{l}{\sqrt{3}}$.
More Information
The radius of gyration is an important concept in mechanics as it helps in understanding how mass is distributed with respect to an axis. The value $\frac{l}{\sqrt{3}}$ indicates that the radius of gyration is less than the length of the rod, reflecting the distribution of mass when the axis of rotation differs from the object's center of mass.
Tips
- Confusing the formula for the moment of inertia for different geometries. Always ensure you are using the correct formula for the shape and the axis of rotation.
- Not simplifying the equations properly. Careful algebraic manipulation is crucial to reach the final answer correctly.
