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Chapter 10: Problem 2

A point on the rim of a rotating wheel has nonzero acceleration, since it's moving in a circular path. Does it necessarily follow that the wheel is undergoing angular acceleration?

Short Answer

Expert verified
No, it does not necessarily follow that the wheel is undergoing angular acceleration. Linear acceleration on a point on a rim of a wheel is present due to the centripetal force that keeps the point moving in a circular path. However, this does not imply that the wheel itself is undergoing angular acceleration.

Step by step solution

01

Understand the Terminologies

It is crucial to understand two kinds of accelerations with respect to a rotating body - Linear and Angular. Linear acceleration is the rate of change of linear speed. Angular acceleration is the rate of change of angular speed. A point on the rim of a rotating wheel is indeed undergoing linear acceleration because it is continually changing its direction.
02

Relate to Centripetal Acceleration

On recognizing that linear acceleration is present because it's moving in a circular path, we see that this is a case of centripetal acceleration - which is the acceleration of an object moving in a circular path. Despite a constant angular speed, the direction of the linear velocity vector changes constantly, indicating acceleration. This is however different from angular acceleration.
03

Define Angular Acceleration

Angular acceleration, on the other hand, refers to how quickly the angular velocity of an object is changing - meaning whether the rate of rotation is increasing or decreasing. This is a different concept from linear acceleration and is not necessarily present when linear acceleration exists.
04

Conclude the Analysis

Examining these principles in relation to one another, it can be concluded that the presence of linear acceleration on a point on the rim of a wheel does not inherently imply that the entire wheel is undergoing angular acceleration. Angular acceleration would only exist if the rate of rotation of the wheel was changing.

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Key Concepts

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

linear acceleration
Linear acceleration is an essential concept when discussing rotational motion. It describes how fast the velocity of an object moving along a straight path is changing. In the context of a rotating wheel, even though the path is curved, every point on the rim experiences a type of linear acceleration. As these points move in a circular path, they continuously change direction, leading to a constant change in velocity. However, linear acceleration does not mean that the wheel must be rotating faster or slower; it only indicates changes in the linear speed of the points on the rim.
  • Linear acceleration = Change in linear speed over time
  • Points in circular motion have linear acceleration due to direction change
  • It doesn't imply change in the actual rate of wheel rotation
angular acceleration
Angular acceleration is the rate at which an object's angular velocity changes with time. It focuses on the rate of rotation of the body itself rather than the linear motion of a point located on the rotating body. The key aspect is that angular acceleration will occur only if there is a change in how fast or slow the object is spinning. For a wheel: if it spins at a constant rate, there is no angular acceleration, even if points on the wheel are accelerating linearly.
  • Angular acceleration 鈮� Linear acceleration
  • Occurs when spinning rate changes 鈥� faster or slower
  • No change in angular velocity = no angular acceleration
centripetal acceleration
Centripetal acceleration is a critical aspect of understanding circular motion. It is directed towards the center of the circular path and is the cause of the continuous change in direction experienced by an object in circular motion. In a rotating wheel, points on the rim feel this acceleration as they move, even when the wheel spins steadily. Centripetal acceleration ensures that the point maintains its curved path rather than moving off in a straight line.
  • Centripetal acceleration points towards circle center
  • Responsible for changing direction of moving points
  • Exists even with constant wheel speed
circular path
A circular path is the trajectory followed by an object moving in a perfect circle, like a point on the rim of a wheel. The presence of linear acceleration in this path demonstrates the interplay between linear and centripetal acceleration. While an object follows this path, gravity or other forces might hold it in circular motion, constantly acting to change its direction and speed. Even if the wheel rotates at a constant angular speed, the circular path ensures continuous movement dynamics.
  • Constantly changing direction due to centripetal forces
  • Interplay between linear and angular dynamics
  • Maintained by forces like tension or gravity
rate of change
Rate of change is a universal concept in kinematics and dynamics that can be applied to both linear and angular motion. This term explains how variables such as speed or velocity change concerning time. For rotating bodies, analyzing rates of change helps predict how motion progresses, whether it's speeding up, slowing down, or maintaining consistent velocity. Grasping these rates is crucial for understanding real-world applications like gears, wheels, and the functionality of various moving parts.
  • Indicates how quickly variables like speed are changing
  • Key to understanding linear and angular motion progression
  • Applicable to real-world machinery and systems

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Most popular questions from this chapter

The shaft connecting a power plant's turbine and electric generator is a solid cylinder of mass \(6.8 \mathrm{Mg}\) and diameter \(85 \mathrm{cm} .\) Find its rotational inertia.

Show that the rotational inertia of a uniform solid spheroid about its axis of revolution is \(\frac{2}{5} M R^{2},\) where \(M\) is its mass and \(R\) is the semi-axis perpendicular to the rotation axis. Why does this result look the same for both a prolate or oblate spheroid and a sphere?

Use integration to show that the rotational inertia of a thick ring of mass \(M\) and inner and outer radii \(R_{1}\) and \(R_{2}\) is given by \(\left.\frac{1}{2} M\left(R_{1}^{2}+R_{2}^{2}\right) . \text { (Hint: See Example } 10.7 .\right)\)

The local historical society has asked your assistance in writing the interpretive material for a display featuring an old steam locomotive. You have information on the torque in a flywheel but need to know the force applied by means of an attached horizontal rod. The rod joins the wheel with a flexible connection \(95 \mathrm{cm}\) from the wheel's axis. The maximum torque the rod produces on the flywheel is \(10.1 \mathrm{kN} \cdot \mathrm{m}\). What force does the rod apply?

Full-circle rotation is common in mechanical systems, but less evident in biology. Yet many single-celled organisms are propelled by spinning, tail-like flagella. The Flagellum of the bacterium \(E\) coli spins at some 600 rad/s, propelling the bacterium at speeds around \(25 \mu \mathrm{m} / \mathrm{s}\). How many revolutions does \(E .\) coli's flagellum make as the bacterium crosses a microscope's \(150-\mu \mathrm{m}-\) wide field of view?
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