Book cover for Physics

Physics

John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler

ISBN #9781118486894

10th Edition

2,562 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces rotational kinematics with an emphasis on the measurement and computation of angular displacement, angular velocity, and angular acceleration. Key relationships such as s = r? for radians, v_T = r? for tangential speed, and a_c = r?² for centripetal acceleration are developed and illustrated through examples like satellite motion, gymnast rotation, and discus throwing. Additionally, the vector nature of angular variables and the importance of unit conversions (especially to radian measure) are highlighted.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

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Example Problems

Example 1

A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of $330 \mathrm{rev} / \mathrm{min}$ (assumed constant) on its way to the catcher's mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

Example 2

The table that follows lists four pairs of initial and final angles of a wheel on a moving car. The elapsed time for each pair of angles is 2.0 s. For each of the four pairs, determine the average angular velocity (magnitude and direction as given by the algebraic sign of your answer). $$\begin{array}{lcc}\hline & \text { Initial angle } \theta_{0} & \text { Final angle } \theta \\\hline \text { (a) } & 0.45 \mathrm{rad} & 0.75 \mathrm{rad} \\\text { (b) } & 0.94 \mathrm{rad} & 0.54 \mathrm{rad} \\\text { (c) } & 5.4 \mathrm{rad} & 4.2 \mathrm{rad} \\\text { (d) } & 3.0 \mathrm{rad} & 3.8 \mathrm{rad} \\\hline\end{array}$$

Example 3

The earth spins on its axis once a day and orbits the sun once a year $\left(365 \frac{1}{4}\right.$ days $)$ Determine the average angular velocity (in $\mathrm{rad} / \mathrm{s}$ ) of the earth as it (a) spins on its axis and (b) orbits the sun. In each case, take the positive direction for the angular displacement to be the direction of the earth's motion.

Example 4

Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is $2.2 \times 10^{20} \mathrm{m},$ and the angular speed of the sun is $1.1 \times 10^{-15} \mathrm{rad} / \mathrm{s}$. How long (in years) does it take for the sun to make one revolution around the center?

Example 5

In Europe, surveyors often measure angles in grads. There are 100 grads in one-quarter of a circle. How many grads are in one radian?
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Step-by-Step Explanations

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Common Mistakes

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