Question

Calculate the moment of inertia of a uniform thin rod of mass \(M\) and length \(l\) about a perpendicular axis of rotation at its end.

Short answer

Question: Calculate the moment of inertia of a uniform thin rod with mass M and length l, about an axis of rotation perpendicular to the rod at one of its ends. Answer: The moment of inertia of a uniform thin rod with mass M and length l about a perpendicular axis of rotation at its end is \(I = \frac{1}{3} Ml^2\).

Step-by-step solution

1. Define the infinitesimal mass element along the rod

Let's denote the length of the rod by \(l\) and its mass by \(M\). We assume the rod has a uniform mass distribution. Let \(x\) be the distance measured along the rod from the end point where the axis of rotation is located. Then, we can define an infinitesimal mass element \(dm\) at position \(x\) on the rod.

2. Determine the relationship between dm and dx

Since the rod has a uniform mass distribution, the mass per unit length \(\lambda\) is constant along the rod. Therefore, we have \(\lambda = \frac{M}{l}\). For the infinitesimal mass element \(dm\), we have: \(dm = \lambda dx = \frac{M}{l} dx\)

3. Set up the definite integral for the moment of inertia

The moment of inertia of an object is given by the integral \(I = \int r^2 dm\), where \(r\) is the perpendicular distance from the axis of rotation to the mass element \(dm\). In this case, \(r\) is equal to \(x\). Thus, the moment of inertia \(I\) can be expressed as: \(I = \int_{0}^{l} x^2 dm\) Using the relationship between \(dm\) and \(dx\), we have: \(I = \int_{0}^{l} x^2 \frac{M}{l} dx\)

4. Perform the integration

Now, we will evaluate the definite integral to find the moment of inertia \(I\): \(I = \frac{M}{l} \int_{0}^{l} x^2 dx\) \(I = \frac{M}{l} \bigg[ \frac{x^3}{3} \bigg]_{0}^{l}\) \(I = \frac{M}{l} \bigg( \frac{l^3}{3} \bigg)\) \(I = \frac{1}{3} Ml^2\) The moment of inertia of a uniform thin rod of mass \(M\) and length \(l\) about a perpendicular axis of rotation at its end is \(\boxed{\frac{1}{3} Ml^2}\).

Key concepts

Uniform Thin Rod

Understanding a uniform thin rod is essential when calculating its moment of inertia. A uniform rod means that its mass is spread out evenly along its entire length. Imagine a long, narrow rod with no parts denser than the rest.
This even distribution of mass allows us to treat the rod as a series of infinitely small mass elements. Each of these elements contributes to the overall moment of inertia.
When dealing with such objects, you often consider:

• Length (\(l\)) - the full distance from one end of the rod to the other.
• Mass (\(M\)) - the total mass spread evenly across the length.
• Mass per Unit Length (\(\lambda\)) - calculated as \(\lambda = \frac{M}{l}\), representing the mass contained in any given unit length of the rod.

This concept is pivotal in setting the stage for more advanced calculations like integrating over the entire rod length to find moments of inertia.

Definite Integral

A definite integral is a mathematical operation that, in this context, helps us cumulate the contributions of each infinitesimal part of the rod to the total moment of inertia. When you break down the rod into small segments, each segment’s contribution is tiny.
By integrating these contributions over the rod's length, you obtain a complete picture (or total inertia).Here’s why it's important:

• Infinitesimal Mass Elements: Connected to differential segments of mass, denoted as \(dm\), involves small lengths (\(dx\)).
• Integral Setup: For an axis perpendicular to one end of the rod, the contribution is calculated using \(I = \int_{0}^{l} x^2 dm\).
• Incorporating Mass Distribution: With \(dm = \frac{M}{l} dx\), this becomes \(I = \int_{0}^{l} x^2 \frac{M}{l} dx\).

The definite integral is evaluated over the limits of the rod's length, yielding a concise solution that details the moment of inertia.

Mass Distribution

Mass distribution plays a crucial role in understanding the physics of a rotating object like a rod. It's about how mass is spread along the rod and how this affects rotational dynamics.
For a uniform thin rod, the mass distribution is uniform, which implies:

• Uniform Distribution: The density \(\lambda = \frac{M}{l}\) remains constant along the rod, meaning the same amount of mass exists in any equally sized segment of the rod.
• Relation to Rotational Inertia: Because the rod is uniform, the calculation becomes simpler and consistent, providing predictable and accurate results.
• Practical Applications: Understanding uniform mass distribution helps when designing systems involving rotational motion, where you need precise control over inertia for stability.

The uniformity ensures that tools like integration return a result reflective of all the rod's properties, making theoretical predictions match real-world observations.