Question

The moment of inertia of a thin rod of mass $\mathrm{M}$ and length $\mathrm{L}$ about an axis passing through the point at a distance $\mathrm{L}/4$ from one of its ends and perpendicular to the rod is $$\begin{gathered} \mathrm{A} \\ \left[\left(7{\mathrm{ML}}^2\right)/48\right] \end{gathered}$$ \{B \} [ $\left[{\mathrm{ML}}^2/12\right]$ $$\begin{gathered} \mathrm{C} \\ [\left({\mathrm{ML}}^2/9\right] \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ [\left({\mathrm{ML}}^2/3\right] \end{gathered}$$

Short answer

The moment of inertia of a thin rod of mass M and length L about an axis passing through the point at a distance L/4 from one of its ends and perpendicular to the rod is: $$\begin{gathered} A \\ \left[\left(7{\mathrm{ML}}^2\right)/48\right] \end{gathered}$$.

Step-by-step solution

Recall the parallel axis theorem

The parallel axis theorem states that the moment of inertia I about any axis can be found using the moment of inertia Ic about a parallel axis passing through the center of mass and the mass M of the object: $$I=I_c+M{d}^2$$ where d is the distance between the two parallel axes.

Calculate the moment of inertia about the center of mass

First, we need to find the moment of inertia of the thin rod about a parallel axis passing through the center of mass. For a thin uniform rod rotating about an axis perpendicular to the rod at its midpoint, the moment of inertia is: $$I_c=\frac{1}{12}M{L}^2$$ We will use this equation in the parallel axis theorem to find the moment of inertia about the given axis.

Calculate the distance between the axes

The problem states that the axis passes through a point at distance L/4 from one end. Since the axis is perpendicular to the rod, and the axis through the center of mass is also perpendicular, we can say that the two axes are parallel to each other. The distance d between the axes is the distance from the center of mass to the given axis, which is: $$d=\frac{L}{4}$$

Use the parallel axis theorem to find the moment of inertia

Now, we use the parallel axis theorem to find the moment of inertia I about the given axis: $$I=I_c+M{d}^2=\frac{1}{12}M{L}^2+M{\left(\frac{L}{4}\right)}^2$$ $$I=\frac{1}{12}M{L}^2+\frac{1}{16}M{L}^2$$ $$I=(\frac{1}{12}+\frac{1}{16})M{L}^2=\frac{7}{48}M{L}^2$$

Compare the result with the given options

We found that the moment of inertia of the thin rod about the given axis is (7/48)ML^2. Comparing this result with the given options, we can see that it matches option A. So the correct answer is: $$\begin{gathered} A \\ \left[\left(7{\mathrm{ML}}^2\right)/48\right] \end{gathered}$$

Key concepts

Parallel Axis Theorem

The Parallel Axis Theorem is an important principle in physics that helps us calculate the moment of inertia of an object when it rotates about an axis that is not aligned with its center of mass. This theorem allows us to find the new moment of inertia (I) by adding the moment of inertia of the object about its center of mass (I_{c}) to an extra term, which accounts for the shift of the rotation axis. The formula is: $$I=I_c+M{d}^2$$Here, $M$ is the mass of the object, and $d$ is the distance between the center of mass axis and the new axis. This theorem is particularly useful when analyzing objects with simple geometries, such as rods, plates, and spheres. By making use of the parallel axis theorem, we can efficiently solve rotational motion problems where the axis of rotation is not through the center of mass.

Thin Rod

A thin rod is a simple geometric object often used in physics problems due to its straightforward shape and uniform mass distribution. In the context of rotational dynamics, understanding the moment of inertia of a thin rod is crucial. For a thin rod, when the axis of rotation passes through its center, the moment of inertia (I_{c}) is given by:$$I_c=\frac{1}{12}M{L}^2$$where $M$ is the mass and $L$ is the length of the rod. This formula makes calculations manageable since it assumes all the mass is evenly spread along the length of the rod. In real-world applications, knowing how to manipulate and apply this formula to various rotational axes is an essential skill.

Rotational Motion

Rotational motion refers to objects moving along a circular path or rotating around an axis. It is a key concept in physics that broadens our understanding of how objects move in the real world. Some essential points to consider in rotational motion include:

• **Angular velocity**: The rate at which an object rotates, typically measured in radians per second.
• **Angular acceleration**: How quickly the rotational speed of an object is changing.
• **Torque**: The force that causes an object to rotate. It is the rotational equivalent of linear force and plays an integral role in understanding dynamics.

For a thin rod in rotational motion, calculating the moment of inertia about different axes is vital, as it affects how quickly or easily the rod can start or stop spinning. Concepts like these are widely applied in various fields such as engineering, astrophysics, and mechanics.

Physics Problem Solving

Solving physics problems effectively requires a systematic approach. The moment of inertia is a classic example of a physics problem that can be solved by following a step-by-step process. Here are some tips to help tackle such problems:

• **Understand the Problem**: Read the problem statement carefully and identify the known and unknown variables.
• **Apply Relevant Theories**: Use foundational theories such as the parallel axis theorem to calculate unknown quantities like the moment of inertia.
• **Break Down Complex Problems**: Simplify the problem into smaller parts, analyze each component, and apply appropriate formulas.
• **Check Units and Dimensions**: Make sure your calculations have consistent units and that the final answer is feasible in terms of physics.

By adhering to this structured approach, you will be better equipped to solve physics problems accurately and efficiently, gaining a deeper insight into the physical principles underlying the problem at hand.