/*! 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 15 Multiple-Concept Example 7 revie... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

Multiple-Concept Example 7 reviews the concepts that play a role in this problem. Car A uses tires for which the coefficient of static friction is 1.1 on a particular unbanked curve. The maximum speed at which the car can negotiate this curve is \(25 \mathrm{m} / \mathrm{s}\). Car \(\mathrm{B}\) uses tires for which the coefficient of static friction is 0.85 on the same curve. What is the maximum speed at which car B can negotiate the curve?

Short Answer

Expert verified
Car B can negotiate the curve at 22.5 m/s.

Step by step solution

01

Understand the Problem

We are given the coefficients of static friction for two cars on the same curve and the maximum speed for car A. We need to find the maximum speed for car B using its coefficient of static friction.
02

Apply the Formula for Maximum Speed

The formula for the maximum speed of a car on an unbanked curve using static friction is given by \( v = \sqrt{\mu_s \cdot g \cdot r} \), where \( \mu_s \) is the coefficient of static friction, \( g \) is the acceleration due to gravity \(9.8 \, \text{m/s}^2\), and \( r \) is the radius of the curve.
03

Calculate the Radius Using Car A's Maximum Speed

Using car A's information, \( v = 25 \, \text{m/s} \) and \( \mu_s = 1.1 \), we can rearrange the formula to find the radius: \( r = \frac{v^2}{\mu_s \cdot g} = \frac{25^2}{1.1 \times 9.8} \approx 57.6 \) meters.
04

Calculate Maximum Speed for Car B

Now that we have the radius of the curve, use car B's coefficient of friction \( \mu_s = 0.85 \) and the same formula to find \( v \): \( v = \sqrt{0.85 \times 9.8 \times 57.6} \).
05

Compute the Value

Calculate \( v = \sqrt{0.85 \times 9.8 \times 57.6} \approx 22.5 \, \text{m/s} \). This is the maximum speed at which car B can negotiate the curve.

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.

Maximum Speed in Circular Motion
When a car navigates a curve, static friction plays a crucial role in determining how fast it can go without slipping. For a car moving in a circle on a flat surface, the maximum speed, where frictional force equals the necessary centripetal force, is key.
The formula \( v = \sqrt{\mu_s \cdot g \cdot r} \) summarizes this relationship, where:
  • \( \mu_s \) is the coefficient of static friction.
  • \( g \) is the gravitational acceleration (~9.8 m/s虏).
  • \( r \) is the curve's radius.
Understanding this equation helps predict how changes in any variable affect maximum speed. For instance, a higher friction coefficient allows faster travel, while a larger radius enables higher speeds due to a gentler curve.
Unbanked Curve Dynamics
Unbanked curves are flat and don't tilt towards the center of the circle. This means that the static friction between the tires and the road is solely responsible for keeping the car moving in a circle.
When the force provided by friction is not sufficient to maintain circular motion, the car may skid outwards. This is why understanding the precise role of static friction is essential in unbanked curve dynamics.
In this scenario, if the curve were banked, the banking angle would add a component of gravitational force to aid in circling the curve, reducing reliance on friction alone. Thus, calculating the frictional limit in unbanked curves ensures safe driving speeds.
Frictional Force Calculations
In physics, frictional force is the resistance force between two surfaces in contact. For circular motion on a flat road, this force is what keeps the car from sliding out of the curve.
The frictional force can be calculated using the formula:
  • \( F_f = \mu_s \cdot N \)
  • Where \( N \) is the normal force, equal to the gravitational force in unbanked scenarios, \( N = m \cdot g \).
By calculating \( F_f \), you can determine how fast the car can go before the frictional force is overcome.
Adjustments in the coefficient of static friction (like using different tires) change the frictional force, influencing the maximum safe speed on the curve. Thus, fine-tuning frictional force through proper calculations and equipment can lead to safer, more controlled driving on curves.

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

Two cars are traveling at the same speed of \(27 \mathrm{m} / \mathrm{s}\) on a curve that has a radius of \(120 \mathrm{m}\). Car \(\mathrm{A}\) has a mass of \(1100 \mathrm{kg}\), and car \(\mathrm{B}\) has a mass of \(1600 \mathrm{kg} .\) Find the magnitude of the centripetal acceleration and the magnitude of the centripetal force for each car.

A satellite is in a circular orbit around an unknown planet. The satellite has a speed of \(1.70 \times 10^{4} \mathrm{m} / \mathrm{s},\) and the radius of the orbit is \(5.25 \times 10^{6} \mathrm{m} .\) A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of \(8.60 \times 10^{6} \mathrm{m} .\) What is the orbital speed of the second satellite?

At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of \(3.2 \mathrm{m} / \mathrm{s},\) and an \(83-\mathrm{kg}\) person feels a \(560-\mathrm{N}\) force pressing against his back. What is the radius of the chamber?

A satellite circles the earth in an orbit whose radius is twice the earth's radius. The earth's mass is \(5.98 \times 10^{24} \mathrm{kg}\), and its radius is \(6.38 \times 10^{6} \mathrm{m}\). What is the period of the satellite?

In an automatic clothes dryer, a hollow cylinder moves the clothes on a vertical circle (radius \(r=0.32 \mathrm{m}),\) as the drawing shows. The appliance is designed so that the clothes tumble gently as they dry. This means that when a piece of clothing reaches an angle of \(\theta\) above the horizontal, it loses contact with the wall of the cylinder and falls onto the clothes below. How many revolutions per second should the cylinder make in order that the clothes lose contact with the wall when \(\theta=70.0^{\circ} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Oscillations

Read Explanation

Nuclear Physics

Read Explanation

Thermodynamics

Read Explanation

Fluids

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.