/*! 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 20 Two banked curves have the same ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 20

Two banked curves have the same radius. Curve \(\mathrm{A}\) 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?

Short Answer

Expert verified
The car can travel around curve B at about 22.34 m/s.

Step by step solution

01

Understanding the Problem

We have two curves with the same radius, each with different banking angles. We are given that a car can travel around curve A without friction at 18 m/s. Our task is to find out the speed at which the same car can travel around curve B under similar conditions.
02

Equation of Motion for Banked Curves

The speed at which a car can travel around a banked curve without friction is given by the formula: \[ v = \sqrt{r \cdot g \cdot \tan{\theta}} \]where \( v \) is the speed, \( r \) is the radius of the curve, \( g \) is the acceleration due to gravity, and \( \theta \) is the banking angle of the curve.
03

Apply Formula to Curve A

For curve A, we know that \( \theta_A = 13^{\circ} \) and \( v_A = 18 \text{ m/s} \). Using the formula, rewrite it as:\[ 18 = \sqrt{r \cdot g \cdot \tan{13^{\circ}}} \]Square both sides to eliminate the square root:\[ 18^2 = r \cdot g \cdot \tan{13^{\circ}} \]
04

Solve for Radius Times Gravity

From the equation obtained from curve A, express \( r \cdot g \) as:\[ r \cdot g = \frac{18^2}{\tan{13^{\circ}}} \]We will use this result to find the speed for curve B.
05

Apply to Curve B Using Known Radius Times Gravity

Considering curve B, we need to find \( v_B \) with \( \theta_B = 19^{\circ} \). Using the motion formula:\[ v_B = \sqrt{r \cdot g \cdot \tan{19^{\circ}}} \]Substitute \( r \cdot g \) from the curve A equation:\[ v_B = \sqrt{\frac{18^2}{\tan{13^{\circ}}} \cdot \tan{19^{\circ}}} \]
06

Calculate the Speed for Curve B

Calculate:1. Evaluate \( \tan{13^{\circ}} \) and \( \tan{19^{\circ}} \).2. Substitute these values to find \( v_B \).\[ v_B = \sqrt{\frac{18^2 \cdot \tan{19^{\circ}}}{\tan{13^{\circ}}}} \approx 22.34 \text{ m/s} \]
07

Verify and Conclude

Verify your calculations to ensure correctness and round-off to appropriate precision if necessary. Therefore, the car can travel around curve B at approximately 22.34 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.

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.

Banked Curves
Banked curves are an essential concept in physics, especially when analyzing circular motion without relying on friction. When a curve is banked, its surface is tilted at a specific angle in relation to the horizontal. This tilt allows vehicles to navigate the curve more safely and efficiently. In a banked curve, the normal force exerted by the surface has a component pointing towards the center of the circle. This central component provides the necessary centripetal force to keep the car on its path. Because of this, when designed correctly, a vehicle can navigate a banked curve without the need for additional friction from the tires.
  • The banking angle ( 胃 ) is crucial. A steeper angle allows higher speeds without sliding.
  • Each angle requires a precise speed for frictionless navigation, dependent on the curve's radius and gravitational pull.
For example, consider a car moving along a curve. At each different banking angle, the suitable speed changes, like how curve B, at a higher angle than curve A, allows for a faster speed without reliance on friction.
Centripetal Force
Centripetal force is the key interplay in maintaining an object鈥檚 circular path, acting towards the center of rotation. When a car travels through a curve, the centripetal force is what keeps it moving along the circular direction rather than in a straight line off the path.For banked curves, this force needs to be achieved without reliance on lateral friction. It is primarily obtained through the horizontal component of the normal force. In physics terms, it can be calculated using the formula:\[ F_c = \frac{mv^2}{r} \]Where:
  • F_c is the centripetal force
  • m is mass
  • v is velocity
  • r is the radius
The beauty of a banked curve is that by carefully selecting the angle and design speed, the centripetal force can be matched exactly, avoiding the need for friction and making navigation smoother and safer.
Trigonometry in Physics
Trigonometry plays a significant role in physics, particularly when dealing with banked curves. The design of these curves relies heavily on understanding and applying trigonometric concepts.One fundamental usage is the tan function, which relates the angle of banking to the speed. The speed of a car in the banked curve formula depends explicitly on the tangent of the banking angle:\[ v = \sqrt{r \cdot g \cdot \tan{\theta}} \]This equation highlights how the tangent function helps calculate the optimal conditions (speed and angle) necessary for a car to traverse a banked curve without sliding due to friction.
  • 胃 represents the angle of banking.
  • The tangent function (\tan{\theta}) conveys how the angle impacts the horizontal force components.
Understanding these relationships is crucial for engineering safe and effective roadways, showcasing the real-world application of trigonometry in solving practical physics problems.

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 car travels at a constant speed around a circular track whose radius is \(2.6 \mathrm{~km} .\) The car goes once around the track in \(360 \mathrm{~s}\). What is the magnitude of the centripetal acceleration of the car?

A 9.5-kg monkey is hanging by one arm from a branch and is swinging on a vertical circle. As an approximation, assume a radial distance of \(85 \mathrm{~cm}\) between the branch and the point where the monkey's mass is located. As the monkey swings through the lowest point on the circle, it has a speed of \(2.8 \mathrm{~m} / \mathrm{s}\). Find (a) the magnitude of the centripetal force acting on the monkey and (b) the magnitude of the tension in the monkey's arm.

There is a clever kitchen gadget for drying lettuce leaves after you wash them. It consists of a cylindrical container mounted so that it can be rotated about its axis by turning a hand crank. The outer wall of the cylinder is perforated with small holes. You put the wet leaves in the container and turn the crank to spin off the water. The radius of the container is 12 \(\mathrm{cm}\). When the cylinder is rotating at 2.0 revolutions per second, what is the magnitude of the centripetal acceleration at the outer wall?

A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bottom of the container. Suppose the centripetal acceleration of the sample is \(6.25 \times 10^{3}\) times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of \(5.00 \mathrm{~cm}\) from the axis of rotation?

A rigid massless rod is rotated about one end in a horizontal circle. There is a mass \(m_{1}\) attached to the center of the rod and a mass \(m_{2}\) attached to the outer end of the rod. The inner section of the rod sustains three times as much tension as the outer section. Find the ratio \(m_{2} / m_{1}\)
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Geometrical and Physical Optics

Read Explanation

Turning Points in Physics

Read Explanation

Nuclear Physics

Read Explanation

Scientific Method Physics

Read Explanation

Kinematics 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.