Problem 61 - Master Resource Book in JEE Main Physics

Question

Since, rod is bent at the middle, so each part of it will ha same length \(\left(\frac{L}{2}\right)\) and \(\operatorname{mass}\left(\frac{M}{2}\right)\) as shown Moment of inertia of each part about an axis passing through its one end $$ =\frac{1}{3}\left(\frac{M}{2}\right)\left(\frac{L}{2}\right)^{2} $$ Hence, net moment of inertia about an axis passing through its middle point \(O\) is $$ I=\frac{1}{3}\left(\frac{M}{2}\right)\left(\frac{L}{2}\right)^{2}+\frac{1}{3}\left(\frac{M}{2}\right)\left(\frac{L}{2}\right)^{2}=\frac{1}{3}\left[\frac{M L^{2}}{8}+\frac{M L^{2}}{8}\right]=\frac{M L^{2}}{12} $$

Step-by-Step Solution

Answer

The net moment of inertia is \( \frac{M L^2}{12} \).

Step 1: Understanding the Rod Configuration

The rod is bent at its midpoint, dividing it into two equal parts. Each part has half the length, \( \frac{L}{2} \), and half the mass, \( \frac{M}{2} \). This setup helps simplify the calculation of the moment of inertia for each section.

Step 2: Calculate Moment of Inertia for Each Part

The moment of inertia of each part of the rod about an axis through one end is calculated using the formula \( \frac{1}{3}mL^2 \). For this divided rod, it becomes: \[ \frac{1}{3} \left(\frac{M}{2}\right) \left(\frac{L}{2}\right)^2 \].

Step 3: Calculate Total Moment of Inertia

The total moment of inertia is the sum of the moment of inertia for both parts since both are equivalent. Thus, the total is: \[ \frac{1}{3} \left(\frac{M}{2}\right) \left(\frac{L}{2}\right)^2 + \frac{1}{3} \left(\frac{M}{2}\right) \left(\frac{L}{2}\right)^2 \].

Step 4: Simplify the Expression

Simplify the total moment of inertia expression to get: \[ \frac{1}{3}\left[\frac{M L^2}{8} + \frac{M L^2}{8}\right] = \frac{1}{3} \times \frac{2M L^2}{8} \]. This simplifies to \( \frac{M L^2}{12} \).

Step 5: Conclusion: Net Moment of Inertia

The net moment of inertia of the bent rod about the axis through its midpoint \( O \) is \( \frac{M L^2}{12} \). This accounts for the contributions from both sections of the rod equally.

Key Concepts

Rotational Dynamics

When diving into rotational dynamics, it's essential to understand the concept of rotational inertia, also known as the moment of inertia. This property reflects how difficult it is to change the rotational motion of a body. Just like mass relates to inertia in linear motion, moment of inertia plays a similar role in rotational motion.
The moment of inertia depends on how the mass is distributed concerning the axis of rotation. Different shapes and mass distributions will result in varying moments of inertia.
For any rotation, think of a wheel for instance, the moment of inertia will determine how quickly or slowly it spins when a force is applied. Objects with a higher moment of inertia require more energy to change their spinning speed.

• The axis through which the object rotates plays a vital role in determining the rotational dynamics.
• Understanding these dynamics involves breaking down complex shapes into simpler ones to calculate the total moment of inertia.

Breaking down a bent rod into segments helps simplify the calculation, which can lead us to the total moment of inertia for more complicated configurations. This is particularly relevant when dealing with rods bent at certain points or complex assemblies of rigid bodies.

Rod Configuration

In our exercise, the rod configuration is split into two equal halves. Each segment of the rod has the same length and mass representation when divided midway. This symmetry simplifies the calculation process.
A rod bent at its midpoint creates two equal sections, each possessing half of the rod's original mass and length. The configuration is crucial to determine how the segments will rotate around a given point, particularly around the center point.

• Each part of the rod behaves individually, and calculating each segment’s moment of inertia allows for summing up to find the total.
• With each section having a known length and mass, it becomes easier to apply standard formulas for the moment of inertia of a rod.

This configuration helps simplify rotational motion calculations by splitting the complex shape into predictable, measurable components. Rod configurations often appear in pendulums, arms of robots, and many mechanical structures where balanced rotation is essential.

Bent Rod Physics

Bent rod physics adds an intriguing subtlety to the calculation of rotational dynamics. In physics, when a rod bends, its axis of symmetry changes, and with it, how we calculate its moment of inertia.
A bent rod behaves differently compared to a straight rod. The bend creates two distinct rotational axes that affect each half of the rod separately. Consider the bent rod as two straight segments rather than a single entity.

• The moment of inertia for each segment is calculated about its own axis before being combined to find the total.
• This kind of setup is frequently found in applications where the distribution of mass influences mechanical stability or function.

Bent rods appear in various engineering applications, from constructing stable bridges to designing high-performance sporting equipment. Understanding bent rod physics is crucial for fields that rely on precise rotational dynamics, like aerospace and automotive engineering.