/*! 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 77 A car of mass \(m\) is being dri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 77

A car of mass \(m\) is being driven on a circular path of radius \(R\). In which of the following circumstances it will not slip \((\mu\) is coefficient of friction between surface and road) (A) \(\frac{m v^{2}}{R} \geq \mu m g\) (B) \(\frac{m v^{2}}{R}=4 \mu m g\) (C) \(\frac{m v^{2}}{R}>m g\) (D) None

Short Answer

Expert verified
The car will not slip under the conditions described by options (A) \(\frac{m v^{2}}{R} \geq \mu m g\) and (B) \(\frac{m v^{2}}{R}=4 \mu m g\).

Step by step solution

01

Write the inequality in terms of force

First, let's write out the forces involved: Static friction force, F_friction = 渭 * m * g Centripetal force, F_centripetal = m * v虏/R For the car to not slip, we need: F_friction >= F_centripetal Which can be written as: 渭 * m * g >= m * v虏/R Now let's analyze each of the provided options.
02

Check option (A)

We will check if the condition of option (A) provides a situation where the car will not slip: \(\frac{m v^{2}}{R} \geq \mu m g\) Comparing this with our inequality, we see that they are the same: 渭 * m * g >= m * v虏/R Thus, option (A) ensures the car will not slip.
03

Check option (B)

We will check if the condition of option (B) provides a situation where the car will not slip: \(\frac{m v^{2}}{R}=4 \mu m g\) Comparing this with the inequality, we get the following condition: \(m\frac{v^2}{R} = 4\mu m g >= \mu m g\) This indicates that the centripetal force acting on the car is greater than or equal to the static friction force in this case, therefore the car will not slip. Hence, option (B) also ensures the car will not slip.
04

Check option (C)

We will check if the condition of option (C) provides a situation where the car will not slip: \(\frac{m v^{2}}{R}>m g\) This condition compares the centripetal force with the gravitational force (m * g) acting on the car. It does not take into account the coefficient of friction 渭. Therefore, this condition does not necessarily imply that the car will not slip, and option (C) is not valid.
05

Check option (D)

Option (D) states "None," which would only be true if none of the options (A), (B), and (C) prevent the car from slipping. However, we have shown that options (A) and (B) do ensure that the car will not slip. Hence, option (D) is not valid. Thus, the conditions under which the car will not slip are described by options (A) and (B).

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.

Coefficient of Friction
The coefficient of friction, usually denoted by \( \mu \), is a measure of how much grip or traction one surface has on another. It plays a crucial role in situations where sliding or slipping might occur, such as a car driving on a circular track. The coefficient expresses the ratio between the force necessary to move one surface over another and the pressure holding them together. There are two types:
  • **Static Friction Coefficient**: This is the friction force that needs to be overcome to start moving an object. It is generally higher than the kinetic coefficient.
  • **Kinetic Friction Coefficient**: Once an object is in motion, this coefficient describes the friction experienced. It is usually lower because less force is needed to keep the object moving.
When a car moves in a circle, the frictional force helps provide the needed grip to prevent sliding outwards. This frictional force can be calculated using the formula: \( F_{\text{friction}} = \mu \cdot m \cdot g \), where \( m \) is the mass of the car, and \( g \) is the acceleration due to gravity. In simple terms, the higher the coefficient of friction, the better the grip.
Centripetal Force
Centripetal force is essential for any object moving in a circular path. It is the inward force required to keep an object moving along a circular trajectory. Imagine you are swinging a ball on a string; the string exerts an inward force called centripetal force that keeps the ball moving in a circle.In the context of a car driving around a circular track, centripetal force is necessary to change the direction of the car's velocity continuously, keeping it on the track. This force is given by the formula:\[F_{\text{centripetal}} = \frac{m v^2}{R}\]where:
  • \( m \) is the mass of the car
  • \( v \) is the velocity or speed at which the car is moving
  • \( R \) is the radius of the circular track
If the centripetal force exceeds the frictional force, the car might lose traction and slide outwards. Therefore, ensuring that frictional force is greater than or equal to the centripetal force is vital for maintaining control, as demonstrated by the conditions in the exercise.
Static Friction
Static friction is the type of friction that comes into play to prevent two surfaces from sliding past each other. It acts as a resistant force when an effort is made to initiate motion between surfaces at rest relative to each other.In our car example, the static friction between the tires and the road provides the gripping force necessary for the car to navigate a circular path without skidding. The strength of this frictional force can be calculated using the equation:\[F_{\text{friction}} = \mu_s \cdot m \cdot g\]Here:
  • \( \mu_s \) is the static friction coefficient
  • \( m \) is the car's mass
  • \( g \) is gravitational acceleration
Static friction must be equal to or surpass the centripetal force to prevent the car from slipping. Unlike kinetic friction, static friction adjusts up to a maximum limit dictated by the coefficient to resist motion. Ensuring that static friction can accommodate the centripetal forces involved is key to safe 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

A particle is rotating in a circle of radius \(R\) with constant angular velocity \(\omega\). Its average velocity during \(t\) seconds after start of motion is (A) \(\frac{2 R}{t} \sin \left(\frac{\omega t}{2}\right)\) (B) \(\frac{2 R}{t} \cos \left(\frac{\omega t}{2}\right)\) (C) \(\frac{R}{t} \sin \left(\frac{\omega t}{2}\right)\) (D) \(\frac{R}{t} \cos \left(\frac{\omega t}{2}\right)\)

A car starts from rest to cover a distance \(x\). The co-efficient of friction between the road and tyres is \(\mu\). The minimum time in which the car can cover distance \(x\) is proportional to (A) \(\mu\) (B) \(\frac{1}{\sqrt{\mu}}\) (C) \(\sqrt{\mu}\) (D) \(\frac{1}{\mu}\)

With what minimum acceleration mass \(M\) must be moved on frictionless surface so that \(m\) remains stick to it as shown. The co-efficient of friction between \(M\) and \(m\) is \(\mu\). (A) \(\mu g\) (B) \(\frac{g}{\mu}\) (C) \(\frac{\mu m g}{M+m}\) (D) \(\frac{\mu m g}{M}\)

A particle of mass \(0.1 \mathrm{~kg}\) is whirled at the end of string in a vertical circle of radius \(1.0 \mathrm{~m}\) at a constant speed of \(5 \mathrm{~m} / \mathrm{s}\). The tension in the string at the highest point of its path is \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(0.5 \mathrm{~N}\) (B) \(1.0 \mathrm{~N} \quad\) (C) \(1.5 \mathrm{~N}\) (D) \(2.0 \mathrm{~N}\)

A constant retarding force of \(50 \mathrm{~N}\) is applied to a body of mass \(20 \mathrm{~kg}\) moving initially with a speed of \(15 \mathrm{~ms}^{-1}\). How long does the body take to stop? (A) \(2 \mathrm{~s}\) (B) \(4 \mathrm{~s}\) (C) \(6 \mathrm{~s}\) (D) \(8 \mathrm{~s}\)
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Scientific Method Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Engineering Physics

Read Explanation

Wave Optics

Read Explanation

Translational Dynamics

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.