/*! 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

In a skating stunt known as crack-the-whip, a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of \(80.0 \mathrm{kg}\) and is \(6.10 \mathrm{m}\) from the pivot. He is skating at a speed of \(6.80 \mathrm{m} / \mathrm{s}\). Determine the magnitude of the centripetal force that acts on him.

Two satellites are in circular orbits around the earth. The orbit for satellite \(\mathrm{A}\) is at a height of \(360 \mathrm{km}\) above the earth's surface, while that for satellite \(\mathrm{B}\) is at a height of \(720 \mathrm{km}\). Find the orbital speed for each satellite.

To create artificial gravity, the space station shown in the drawing is rotating at a rate of 1.00 rpm. The radii of the cylindrically shaped chambers have the ratio \(r_{\lambda} / r_{\mathrm{B}}=4.00 .\) Each chamber A simulates an acceleration due to gravity of \(10.0 \mathrm{m} / \mathrm{s}^{2} .\) Find values for (a) \(r_{\mathrm{A}}\), (b) \(r_{\mathrm{B}}\), and (c) the acceleration due to gravity that is simulated in chamber B.

Two banked curves have the same radius. Curve A is banked at an angle of \(13^{\circ},\) and curve \(\mathrm{B}\) is banked at an angle of \(19^{\circ} .\) A car can travel around curve A without relying on friction at a speed of \(18 \mathrm{m} / \mathrm{s}\). At what speed can this car travel around curve \(\mathrm{B}\) without relying on friction?

Multiple-Concept Example 7 deals with the concepts that are important in this problem. A penny is placed at the outer edge of a disk (radius \(=0.150 \mathrm{m})\) that rotates about an axis perpendicular to the plane of the disk at its center. The period of the rotation is 1.80 s. Find the minimum coefficient of friction necessary to allow the penny to rotate along with the disk.
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Quantum Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Rotational Dynamics

Read Explanation

Physical Quantities And Units

Read Explanation

Energy 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.