/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 A thin rod has a length of \(0.2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 63

A thin rod has a length of \(0.25 \mathrm{m}\) and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of \(0.32 \mathrm{rad} / \mathrm{s}\) and a moment of inertia of \(1.1 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2} .\) A bug standing on the axis decides to crawl out to the other end of the rod. When the bug (mass \(=4.2 \times 10^{-3} \mathrm{kg}\) ) gets where it's going, what is the angular velocity of the rod?

Short Answer

Expert verified
The final angular velocity of the rod is about 0.20 rad/s.

Step by step solution

01

Understand the conservation principle

Since no external torques act on the system, the principle of conservation of angular momentum applies. This principle states that the initial angular momentum of the system must equal the final angular momentum. Therefore, we have \( L_i = L_f \), where \( L \) represents angular momentum.
02

Calculate initial angular momentum

The initial angular momentum \( L_i \) of the system can be calculated using the formula \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. Substituting the given values, \( L_i = (1.1 \times 10^{-3} \, \text{kg} \cdot \text{m}^2)(0.32 \, \text{rad/s}) \). Calculate this value.
03

Calculate the final moment of inertia

When the bug reaches the end of the rod, it adds its own moment of inertia, which is calculated as \( I_{bug} = m r^2 \). Here, \( m = 4.2 \times 10^{-3} \, \text{kg} \) and \( r = 0.25 \, \text{m} \) (the length of the rod). Therefore, \( I_{bug} = (4.2 \times 10^{-3})(0.25^2) \). Calculate this value.
04

Find total final moment of inertia

Add the moment of inertia of the rod and the moment of inertia of the bug to get the total final moment of inertia: \( I_f = I_{rod} + I_{bug} \). Use the previously calculated \( I_{bug} \) and given \( I_{rod} = 1.1 \times 10^{-3} \, \text{kg}\cdot\text{m}^2 \).
05

Calculate final angular velocity

Using the conservation of angular momentum, set \( L_i = L_f \). We have the equation: \( I_i \omega_i = I_f \omega_f \). Solve for the final angular velocity \( \omega_f \): \( \omega_f = \frac{I_i \omega_i}{I_f} \). Substitute the known values to find \( \omega_f \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Angular Velocity
Angular velocity is a key concept in rotational motion, representing how fast an object rotates around an axis. It is denoted by the Greek letter \( \omega \) and is measured in radians per second (rad/s). Imagine a merry-go-round. The faster it spins, the higher its angular velocity. Unlike linear velocity, which deals with straight-line motion, angular velocity concerns circular paths. It's about the rate at which something spins around its center or pivot point.

To relate angular velocity to our problem: the thin rod initially rotates with an angular velocity of \(0.32 \, \text{rad/s}\). This speed is crucial because it helps us calculate the initial angular momentum of the system. Angular velocity changes if the distribution of mass in the rotating object changes, as seen when the bug crawls to the end of the rod, affecting the system's overall moment of inertia.
  • Importance: Angular velocity is central to understanding rotational motion dynamics.
  • Key usage: Used to calculate how rotation speed varies with changing mass distribution.
Explaining the Moment of Inertia
Moment of inertia is akin to mass in linear motion but applies to rotational motion. It measures an object's resistance to changes in its rotational speed. Think about it as the rotational "heaviness"; a wider or heavier object is harder to spin. This concept is crucial when discussing objects like wheels or rotating rods, as in our exercise.

In mathematical terms, the moment of inertia \( I \) depends on the distribution of mass relative to the axis of rotation. For the thin rod in the exercise, the moment of inertia is provided as \(1.1 \times 10^{-3} \, \text{kg} \cdot \text{m}^2\). When the bug moves along with the rod, it adds its own moment of inertia, calculated using \(I_{\text{bug}} = m r^2\), where \(m\) is the bug's mass, and \(r\) is the distance from the axis.
  • Role: Moment of inertia determines how mass distribution affects rotation.
  • Effect: Changes in moment of inertia affect angular velocity due to conservation laws.
Rotational Motion and Its Dynamics
Rotational motion involves objects spinning around an axis and is a common phenomenon in physics. Unlike linear motion, where objects move along a path, rotational motion involves circular paths. It is governed by principles like angular velocity and moment of inertia, which we have already discussed.

An essential concept linked with rotational motion is the conservation of angular momentum. This principle states that if no external torque acts upon a system, the total angular momentum remains constant. In our exercise, the initial angular momentum of the rod system equals the final angular momentum, even after the bug changes its position on the rod. This is formulated as \( L_i = L_f \), where \( L \) represents angular momentum.
  • Dynamic essence: Rotational motion behavior is dictated by angular velocity and moment of inertia.
  • Conservation: Allows us to predict rotational behavior when mass distribution changes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Calculate the kinetic energy that the earth has because of (a) its rotation about its own axis and (b) its motion around the sun. Assume that the earth is a uniform sphere and that its path around the sun is circular. For comparison, the total energy used in the United States in one year is about \(1.1 \times 10^{20} \mathrm{J}\)

A solid disk rotates in the horizontal plane at an angular velocity of \(0.067 \mathrm{rad} / \mathrm{s}\) with respect to an axis perpendicular to the disk at its center. The moment of inertia of the disk is \(0.10 \mathrm{kg} \cdot \mathrm{m}^{2}\). From above, sand is dropped straight down onto this rotating disk, so that a thin uniform ring of sand is formed at a distance of \(0.40 \mathrm{m}\) from the axis. The sand in the ring has a mass of \(0.50 \mathrm{kg} .\) After all the sand is in place, what is the angular velocity of the disk?

Two thin rods of length \(L\) are rotating with the same angular speed \(\omega\) (in \(\mathrm{rad} / \mathrm{s}\) ) about axes that pass perpendicularly through one end. Rod \(\mathrm{A}\) is massless but has a particle of mass \(0.66 \mathrm{kg}\) attached to its free end. Rod B has a mass of 0.66 kg, which is distributed uniformly along its length. The length of each rod is \(0.75 \mathrm{m},\) and the angular speed is \(4.2 \mathrm{rad} / \mathrm{s}\). Find the kinetic energies of rod \(A\) with its attached particle and of rod \(B\).

A person is standing on a level floor. His head, upper torso, arms, and hands together weigh \(438 \mathrm{N}\) and have a center of gravity that is \(1.28 \mathrm{m}\) above the floor. His upper legs weigh \(144 \mathrm{N}\) and have a center of gravity that is \(0.760 \mathrm{m}\) above the floor. Finally, his lower legs and feet together weigh \(87 \mathrm{N}\) and have a center of gravity that is \(0.250 \mathrm{m}\) above the floor. Relative to the floor, find the location of the center of gravity for his entire body.

Starting from rest, a basketball rolls from the top of a hill to the bottom, reaching a translational speed of \(6.6 \mathrm{m} / \mathrm{s} .\) Ignore frictional losses. (a) What is the height of the hill? (b) Released from rest at the same height, a can of frozen juice rolls to the bottom of the same hill. What is the translational speed of the frozen juice can when it reaches the bottom?
See all solutions

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Read Explanation

Physics of Motion

Read Explanation

Medical Physics

Read Explanation

Modern Physics

Read Explanation

Electromagnetism

Read Explanation

Famous Physicists

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.