Problem 46 - Thomas Calculus

Question

Radius of gyration of slender rod Find the radius of gyration of a slender rod of constant linear density \(\delta \mathrm{gm} / \mathrm{cm}\) and length \(L\) \(\mathrm{cm}\) with respect to an axis a. through the rod's center of mass perpendicular to the rod's axis. b. perpendicular to the rod's axis at one end of the rod.

Step-by-Step Solution

Answer

a. \( \frac{L}{\sqrt{12}} \); b. \( \frac{L}{\sqrt{3}} \).

Step 1: Understanding Radius of Gyration

The radius of gyration, denoted as \( k \), is a measure that describes how the mass of an object is distributed with respect to an axis. It is calculated using the moment of inertia \( I \) and the object's total mass \( M \) by the formula \( k = \sqrt{\frac{I}{M}} \).

Step 2: Calculate Linear Mass Density

The linear mass density of the rod is given as \( \delta \mathrm{gm}/\mathrm{cm} \). This value describes how much mass there is per unit length of the rod.

Step 3: Determine Total Mass of the Rod

Since the rod has a constant linear density \( \delta \) and a total length \( L \), the total mass of the rod \( M \) is calculated as \( M = \delta \times L \).

Step 4: Moment of Inertia Around Center of Mass

For a slender rod rotating about an axis through its center of mass (and perpendicular to its length), the moment of inertia is given by \( I_{cm} = \frac{1}{12}ML^2 \).

Step 5: Radius of Gyration Around Center of Mass

Substitute \( I_{cm} = \frac{1}{12}ML^2 \) and \( M = \delta L \) into the formula for radius of gyration: \( k_{cm} = \sqrt{\frac{\frac{1}{12}ML^2}{M}} = \sqrt{\frac{L^2}{12}} = \frac{L}{\sqrt{12}} \).

Step 6: Moment of Inertia Around End of Rod

For a slender rod rotating about an axis through one end and perpendicular to its length, the moment of inertia is given by \( I_{end} = \frac{1}{3}ML^2 \).

Step 7: Radius of Gyration Around End of Rod

Substitute \( I_{end} = \frac{1}{3}ML^2 \) and \( M = \delta L \) into the formula for radius of gyration: \( k_{end} = \sqrt{\frac{\frac{1}{3}ML^2}{M}} = \sqrt{\frac{L^2}{3}} = \frac{L}{\sqrt{3}} \).

Key Concepts

Moment of Inertia

In simple terms, the moment of inertia is like mass for rotation. It tells us how difficult it is to spin an object around a specific axis. - Imagine trying to spin a thin rod. The way the mass spreads out along the rod affects how easy or hard it is to spin. - The moment of inertia depends on both the mass of the object and how that mass is spread out. The further the mass is from the axis of rotation, the larger the moment of inertia. For a slender rod rotating about an axis through its center (like when you flip a stick), the moment of inertia is given by: \[ I_{cm} = \frac{1}{12}ML^2 \] This formula shows that the moment of inertia depends on the length of the rod squared. It means that if you make the rod twice as long, the moment of inertia will increase by four times if the mass remains constant.

Linear Density

Linear density is a measure of how much mass lies along a unit length of an object. Think of it as how tightly packed the mass is along the length. For our slender rod, we have: - Linear density, denoted by \( \delta \), simply measures mass per unit length (e.g., grams per centimeter). - Since the linear density is constant for a slender rod, we can easily determine the total mass by multiplying the linear density by the length of the rod: \[ M = \delta \times L \] This relationship helps us calculate the mass of the rod quickly and directly, simply by knowing its length and linear density, which is particularly useful in engineering and physics problems.

Center of Mass

The center of mass is like the 'balancing point' of an object. It's the point where the mass is considered to be concentrated for the purpose of analyzing translational motion. - For a uniformly dense rod, the center of mass is exactly at the midpoint. - Balancing a rod on a finger and finding that sweet spot – that's you finding the center of mass! In our exercise, the center of mass serves as a key reference point for calculating the moment of inertia. When the axis of rotation passes through this point, the calculation for moment of inertia becomes: \[ I_{cm} = \frac{1}{12} ML^2 \] This calculation is crucial for determining how an object will behave when rotating.