Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 68

Two gears \(A\) and \(B\) are in contact. The radius of gear \(A\) is twice that of gear \(B\). (a) When \(A\) 's angular velocity is \(6.00 \mathrm{Hz}\) counterclockwise, what is \(B^{\prime}\) s angular velocity? (b) If \(A\) 's radius to the tip of the teeth is \(10.0 \mathrm{cm},\) what is the linear speed of a point on the tip of a gear tooth? What is the linear speed of a point on the tip of \(B\) 's gear tooth?

Short Answer

Expert verified
Answer: The linear speed of a point on the tip of Gear B's gear tooth is \(1.2\pi \mathrm{m/s}\).

Step by step solution

01

Understand the relationship between gears A and B and find a ratio

Since the radius of gear A is twice that of gear B, their circumferences have the same ratio because the circumference is directly proportional to the radius. The ratio is 2:1 and since the gears are in contact, they must have the same linear velocity.
02

Calculate angular velocity of gear B

Angular velocity of gear A (\(\omega_A\)) is given as \(6.00 \mathrm{Hz}\). To find the angular velocity of gear B (\(\omega_B\)), we divide gear A's angular velocity by the ratio (2:1) found in the previous step. $$ \omega_B = \frac{6.00 \mathrm{Hz}}{\frac{2}{1}} = 3.00 \mathrm{Hz} $$
03

Calculate the linear speed of the tip of Gear A

Now that we found the angular velocity of Gear A, we can calculate the linear speed (\(v_A\)) of a point on the tip of the gear tooth. We use the following equation: $$ v_A = r_A \cdot \omega_A $$ where, \(r_A\) is the radius of Gear A, and \(\omega_A = 6.00 \mathrm{Hz}\) Given Gear A's radius is \(10.0 \mathrm{cm}\) or \(0.1 \mathrm{m}\), we can calculate the linear speed: $$ v_A = 0.1 \mathrm{m} \cdot 6.00 \text{ 2}\pi \mathrm{rad/s} = 1.2\pi \mathrm{m/s} $$
04

Calculate the linear speed of the tip of Gear B

Since gears A and B are in contact and have the same linear velocity (as found in step 1), the linear speed of a point on the tip of Gear B's gear tooth is equal to the linear speed of Gear A which is \(1.2\pi \mathrm{m/s}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 soccer ball of diameter \(31 \mathrm{cm}\) rolls without slipping at a linear speed of \(2.8 \mathrm{m} / \mathrm{s} .\) Through how many revolutions has the soccer ball turned as it moves a linear distance of \(18 \mathrm{m} ?\)
A small body of mass \(0.50 \mathrm{kg}\) is attached by a \(0.50-\mathrm{m}-\) long cord to a pin set into the surface of a frictionless table top. The body moves in a circle on the horizontal surface with a speed of $2.0 \pi \mathrm{m} / \mathrm{s} .$ (a) What is the magnitude of the radial acceleration of the body? (b) What is the tension in the cord?
A spacecraft is in orbit around Jupiter. The radius of the orbit is 3.0 times the radius of Jupiter (which is \(\left.R_{1}=71500 \mathrm{km}\right) .\) The gravitational field at the surface of Jupiter is 23 N/kg. What is the period of the spacecraft's orbit? [Hint: You don't need to look up any more data about Jupiter to solve the problem.]
The time to sunset can be estimated by holding out your arm, holding your fingers horizontally in front of your eyes, and counting the number of fingers that fit between the horizon and the setting Sun. (a) What is the angular speed, in radians per second, of the Sun's apparent circular motion around the Earth? (b) Estimate the angle subtended by one finger held at arm's length. (c) How long in minutes does it take the Sun to "move" through this same angle?
A carnival swing is fixed on the end of an 8.0 -m-long beam. If the swing and beam sweep through an angle of \(120^{\circ},\) what is the distance through which the riders move?
See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Kinematics Physics

Read Explanation

Electromagnetism

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Energy Physics

Read Explanation

Wave Optics

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.

Sign-up for free