/*! 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 3 ssm A pitcher throws a curveball... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 3

ssm A pitcher throws a curveball that reaches the catcher in \(0.60 \mathrm{~s}\). The ball curves because it is spinning at an average angular velocity of 330 rev/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?

Short Answer

Expert verified
The angular displacement is 20.736 radians.

Step by step solution

01

Convert Angular Velocity to Radians per Second

First, convert the angular velocity from revolutions per minute (rev/min) to radians per second (rad/s). There are \(2\pi\) radians in one revolution and 60 seconds in one minute, so:\[\omega = 330 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{\text{rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 34.56 \frac{\text{rad}}{\text{s}}.\]
02

Use Angular Displacement Formula

The formula for angular displacement is \(\theta = \omega \cdot t\), where \(\omega\) is the angular velocity in rad/s, and \(t\) is the time in seconds. Given \(t = 0.60\) s and \(\omega = 34.56\) rad/s:\[\theta = 34.56 \frac{\text{rad}}{\text{s}} \times 0.60 \text{ s} = 20.736 \text{ rad}.\]
03

Final Calculation and Result Interpretation

Calculate the angular displacement \(\theta = 20.736\) rad. This means that as the baseball travels from the pitcher to the catcher, it spins through an angular displacement of 20.736 radians.

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.

Angular Velocity
Angular velocity refers to how fast something spins around an axis. In physics, it's often represented by the symbol \(\omega\). For example, a spinning baseball has angular velocity, indicating how many revolutions it makes over time. Usually, we measure this in units like revolutions per minute (rev/min) or radians per second (rad/s).

To understand angular velocity better, think of a spinning record on a turntable. The speed of its spin as it rotates, given in a suitable unit of measurement, indicates its angular velocity.
  • Unit Conversion: Sometimes, we need to convert from one unit to another, such as rev/min to rad/s, which involves using the relationship that one revolution equals \(2\pi\) radians and one minute equals 60 seconds.
  • Practical Example: A baseball spinning with an angular velocity of 330 rev/min needs conversion to rad/s, essential for calculating further properties like angular displacement.
Radians
Radians are a unit of angular measurement used in math and physics to measure angles. Unlike degrees, which divide a circle into 360 parts, radians use the circle's own radius for measurement, defining it more naturally in terms of the arc.

One full turn (or revolution) around a circle is equal to \(2\pi\) radians. This makes the radian a very convenient unit when working with circular motion, like the spinning of a baseball.

Why do we use radians?
  • They simplify many mathematical equations. For instance, the arc length of a circle can directly be calculated as \(\text{arc length} = \theta \times \text{radius}\).
  • In trigonometry functions and equations, radians provide a simpler and more direct measurement than degrees.
Understanding radians is crucial when dealing with angular velocity and displacement, both of which often use radians as the unit of measure.
Angular Motion
Angular motion describes the movement path of objects that rotate or spin around a central point or axis. Unlike linear motion, which moves in a straight line, angular motion involves rotation, which can be observed in systems like spinning wheels or orbiting planets.

In our example with the baseball, its motion includes angular velocity and angular displacement, both key features describing how the ball spins as it moves towards the catcher.
  • Angular Velocity: This specifies the rate of rotation, measured in rad/s, and tells us how fast the baseball spins.
  • Angular Displacement: This refers to the angle through which an object has rotated or moved, represented in radians.
Understanding angular motion involves calculating these parameters to fully grasp how objects behave when they rotate. These calculations help in determining effects like curvature of path and amount of rotation.
Conversion of Units
In physics and mathematics, conversion of units is vital for problem-solving and understanding measurements. Often, calculations require converting from one unit to another to match the given formulas or to comprehend the problem's context.

For angular measurements:
  • From Revolutions to Radians: Since one revolution equals \(2\pi\) radians, we multiply the number of revolutions by \(2\pi\) to convert.
  • From Minutes to Seconds: As long as a unit of time is involved, ensure a coherent unit by converting minutes into seconds, using the factor of 60 seconds per minute.
In our exercise, we converted angular velocity from rev/min to rad/s because the angular displacement formula \(\theta = \omega \cdot t\) requires \(\omega\) in rad/s. Learning these conversions ensures precision and helps avoid mistakes during calculations. Understanding these unit conversions is a foundational skill in physics and engineering.

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

A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of \(64 \mathrm{~cm}\) and is wound around the top at a spot where its radius is \(2.0 \mathrm{~cm}\). The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of \(+12 \mathrm{rad} / \mathrm{s}^{2}\). What is the final angular velocity of the top when the string is completely unwound?

The earth spins on its axis once a day and orbits the sun once a year \(\left(365^{1 / 4}\right.\) days). Determine the average angular velocity (in rad/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.

Does the tip of a rotating fan blade have a tangential acceleration when the blade is rotating (a) at a constant angular velocity and (b) at a constant angular acceleration? Provide reasons for your answers. Problem A fan blade is rotating with a constant angular acceleration of \(+12.0 \mathrm{rad} / \mathrm{s}^{2}\). At what point on the blade, measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity?

ssm An electric circular saw is designed to reach its final angular speed, starting from rest, in \(1.50 \mathrm{~s}\). Its average angular acceleration is \(328 \mathrm{rad} / \mathrm{s}^{2}\). Obtain its final angular speed.

A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of \(-4.00 \mathrm{rad} / \mathrm{s}^{2}\). Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of \(-25.0 \mathrm{rad} / \mathrm{s}\). While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Atoms and Radioactivity

Read Explanation

Electricity and Magnetism

Read Explanation

Fluids

Read Explanation

Thermodynamics

Read Explanation

Waves Physics

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.