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Chapter 9: Problem 32

Three small blocks, each with mass \(m,\) are clamped at the ends and at the center of a rod of length \(L\) and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod and (b) a point onefourth of the length from one end.

Short Answer

Expert verified
The moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod is \(\frac{mL^2}{2}\) and (b) a point one-fourth of the length from one end is \(\frac{11mL^2}{16}\).

Step by step solution

01

Determine Position of Blocks for Part (a)

The rod has three blocks: two at each end and one at the center. Since the rod has negligible mass, we only consider the blocks for the moment of inertia. For part a, the axis of rotation is at the center of the rod. Thus, the distance for the block at the center is 0, and the distance for the two blocks at the ends is \(L/2\) each.
02

Calculate Moment of Inertia for Part (a)

The moment of inertia for part a is calculated using the formula \(I = \sum m_i r_i^2\). The two blocks at the ends each have a distance of \(L/2\) from the axis of rotation, while the block at the center has distance 0. Thus, we have \(I = m(0^2) + m(\frac{L}{2})^2 + m(\frac{L}{2})^2 = m(\frac{L}{2})^2 + m(\frac{L}{2})^2 = m(\frac{L^2}{4}) + m(\frac{L^2}{4}) = \frac{mL^2}{2}\).
03

Determine Position of Blocks for Part (b)

For part b, the axis of rotation is one fourth of the length from one end. Thus, the distances from the axis for the three blocks are \(L/4\), \(L/4\) and \(3L/4\).
04

Calculate Moment of Inertia for Part (b)

The moment of inertia for part b is calculated using the formula \(I = \sum m_i r_i^2\). For the three blocks, this becomes \(I = m(\frac{L}{4})^2 + m(\frac{L}{4})^2 + m(\frac{3L}{4})^2 = \frac{mL^2}{16} + \frac{mL^2}{16} + \frac{9mL^2}{16} = \frac{11mL^2}{16}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics
Physics serves as the foundation of numerous concepts in science and engineering, chiefly concerning the nature, properties, and behaviors of matter and energy. Central to physics is the analysis of forces and their impact on matter, all of which is essential for understanding the dynamics of rotating systems.

In the context of rotational motion, physics provides the principles through which we analyze the tendency of objects to resist changes in their motion, a concept known as inertia. This is particularly relevant when studying systems composed of multiple masses in rotation, as in the exercise with the three blocks and the rod.
Rotational Motion
Rotational motion refers to the movement of a body about an axis. This axis could either pass through the body or lie outside of it. Objects rotating about an axis demonstrate characteristics such as angular velocity, angular acceleration, and a unique property known as moment of inertia.

Understanding how mass distribution affects an object's resistance to rotational acceleration is the key to solving problems involving rotational motion, much like the textbook exercise where blocks are placed at different points along a rod.
Moment of Inertia Calculation
The moment of inertia, symbolized by 'I', is a measure of an object's resistance to change in its rotational motion. It signifies how the mass of a body is distributed relative to the axis of rotation. For a single point mass, the moment of inertia is calculated by the formula, \( I = mr^2 \), where 'm' represents mass and 'r' is the distance from the mass to the axis of rotation.

For a system with multiple masses, such as in the exercise, the total moment of inertia can be found by summing the individual moments of inertia: \( I = \sum m_ir_i^2 \). This formula was applied to the given rod with three blocks to calculate its moment of inertia about two different axes, demonstrating that moment of inertia depends not only on the mass of the objects but also on their positions relative to the axis.
Axis of Rotation
The axis of rotation is an imaginary line around which a body rotates. In the scenario presented in the textbook exercise, we considered two different axes: one through the center of the rod and another one a quarter of its length from one end. The position of this axis plays a vital role in determining the moment of inertia of a system.

Changing the axis of rotation, as seen between parts (a) and (b) of the exercise, results in different values of the moment of inertia for the same collection of masses. The further the mass is from the axis, the greater its contribution to the moment of inertia, thus affecting the overall rotational dynamics of the system.

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Most popular questions from this chapter

A circular saw blade with radius \(0.120 \mathrm{~m}\) starts from rest and turns in a vertical plane with a constant angular acceleration of \(2.00 \mathrm{rev} / \mathrm{s}^{2} .\) After the blade has turned through \(155 \mathrm{rev},\) a small piece of the blade breaks loose from the top of the blade. After the piece breaks loose, it travels with a velocity that is initially horizontal and equal to the tangential velocity of the rim of the blade. The piece travels a vertical distance of \(0.820 \mathrm{~m}\) to the floor. How far does the piece travel horizontally, from where it broke off the blade until it strikes the floor?

A new species of eel is found to have the same mass but onequarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has (a) half the moment of inertia as the American eel; (b) the same moment of inertia as the American eel; (c) twice the moment of inertia as the American eel; (d) four times the moment of inertia as the American eel.

A bucket of mass \(m\) is tied to a massless cable that is wrapped around the outer rim of a uniform pulley of radius \(R,\) on a frictionless axle, similar to the system shown in Fig. E9.47. In terms of the stated variables, what must be the moment of inertia of the pulley so that it always has half as much kinetic energy as the bucket?

A thin uniform rod of mass \(M\) and length \(L\) is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the midpoint of the line connecting its two ends.

A pulley on a frictionless axle has the shape of a uniform solid disk of mass \(2.50 \mathrm{~kg}\) and radius \(20.0 \mathrm{~cm}\). A \(1.50 \mathrm{~kg}\) stone is attached to a very light wire that is wrapped around the rim of the pulley (Fig. E9.47), and the system is released from rest. (a) How far must the stone fall so that the pulley has \(4.50 \mathrm{~J}\) of kinetic energy? (b) What percent of the total kinetic energy does the pulley have?
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