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Chapter 4: Problem 69

A car starts from rest to cover a distance \(s\). The coefficient of friction between the road and the tyres is \(\mu\). The minimum time in which the car can cover the distance is proportional to: (a) \(\boldsymbol{\mu}\) (b) \(\sqrt{\mu}\) (c) \(1 / \mu\) (d) \(1 / \sqrt{\mu}\)

Short Answer

Expert verified
The minimum time is proportional to \( 1/\sqrt{\mu} \), so option (d) is correct.

Step by step solution

01

Understand the Question

We need to determine how the minimum time required for a car to cover a distance relates to the coefficient of friction \( \mu \) between the tyres and the road.
02

Relate Variables Using Physics Principles

The frictional force \( f \) is equal to \( \mu \times N \), where \( N \) is the normal force. Since the car is on a flat road, \( N = mg \), where \( m \) is the mass of the car and \( g \) is the acceleration due to gravity. Thus, the maximum force is \( f = \mu mg \). This force provides the maximum acceleration \( a \), so \( a = \frac{\mu mg}{m} = \mu g \).
03

Use Kinematic Equations

For a car starting from rest, we can use the kinematic equation \( s = \frac{1}{2} at^2 \) to find time \( t \). Rearranging gives \( t = \sqrt{\frac{2s}{a}} \). Substituting \( a = \mu g \) into the equation we get \( t = \sqrt{\frac{2s}{\mu g}} \).
04

Determine Proportional Relationship

The expression for time \( t \) shows it is proportional to \( \frac{1}{\sqrt{\mu}} \). Specifically, \( t \) contains \( \frac{1}{\sqrt{\mu}} \) within the square root. Therefore, the minimum time is proportional to \( \frac{1}{\sqrt{\mu}} \).
05

Choose the Correct Option

Based on the relationship determined, the correct option that matches the proportional relationship is (d) \( 1 / \sqrt{\mu} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Friction
Friction is a force that opposes the motion of an object in contact with a surface. Imagine trying to slide a book across a table. The book doesn't move on its own because friction is opposing the push. The coefficient of friction, denoted as \( \mu \), is a measure of how much frictional force is present between two surfaces.
This value depends on the materials in contact: rubber on asphalt has high friction (a high \( \mu \)), while ice on steel has low friction (a low \( \mu \)).
The frictional force \( f \) can be calculated by multiplying the coefficient of friction \( \mu \) by the normal force \( N \) acting on the object. On a flat surface, the normal force is usually the object's weight \( mg \), where \( m \) is mass and \( g \) is gravitational acceleration. Therefore, the force equation becomes \( f = \mu mg \).
  • Friction opposes motion.
  • \( \mu \) is a unitless value that varies based on material contact.
  • The frictional force formula is \( f = \mu N \).
Acceleration
Acceleration refers to how quickly an object speeds up or slows down. If you're in a car that suddenly speeds up to join the highway, you feel pushed back into your seat鈥攖his is due to acceleration. In physics, acceleration \( a \) is often calculated using forces acting on an object.
For a car on a flat road, the frictional force helps in accelerating the car forward by converting friction into motion.
Given that frictional force \( f \) is \( \mu mg \), we derive the formula for acceleration as \( a = \frac{f}{m} = \mu g \).
  • Acceleration describes changes in speed.
  • It is dependent on the driving force鈥攊n this case, friction.
  • In our exercise equation, it results in \( a = \mu g \).
Kinematic Equations
Kinematic equations help us describe motion in physics. When an object starts from rest, these equations can tell us how far it will travel over time or how long it takes to reach a certain speed.
One important equation in kinematics, particularly for objects accelerating from a standstill, is \( s = \frac{1}{2} at^2 \), where \( s \) is the distance covered, \( a \) is acceleration, and \( t \) is time.
If you want to calculate the time it takes for a car to cover a distance \( s \) given an acceleration \( a = \mu g \), rearrange the equation to find \( t = \sqrt{\frac{2s}{a}} \).
To simplify using our conditions, substitute \( a = \mu g \) into it, giving \( t = \sqrt{\frac{2s}{\mu g}} \).
  • Kinematic equations predict an object's motion.
  • They relate distance, speed, acceleration, and time.
  • Helps find time \( t \) required given frictional effects on acceleration.

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Most popular questions from this chapter

A bicycle is in motion. When it is not pedaled, the force of friction exerted by the ground on the two wheels is such that it acts: (a) in the backward direction on the front wheel and in the forward direction on the rear wheel (b) in the forward direction on the front wheel and in the backward direction on rear wheel (c) in the backward direction on both the front and the rear wheels (d) in the forward direction on both the front and the rear wheels

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that: (a) its velocity is constant (b) its kinetic energy is constant (c) it moves in a circular path (d) both (b) and (c) are correct

For the equilibrium of a body on an inclined plane of inclination \(45^{\circ}\), the coefficient of static friction will be : (a) greater than one (b) less than one (c) zero (d) less than zero

Two blocks \(A\) and \(B\) of masses \(4 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) respectively rest on a smooth horizontal surface. The coefficient of friction between \(A\) and \(B\) is \(0.36\). Then : (a) the maximum horizontal force \(F\) which can be applied to \(B\) so that there is no relative motion between \(A\) and \(B\) is equal to \(0.36 \times 3 \times 9.8 \mathrm{~N}\) (b) the maximum horizontal force \(F\) on \(B\) with no relative motion between \(A\) and \(B\) is equal to \(0.63 \times 3 \times 9.8 \mathrm{~N}\) (c) the maximum horizonta' force \(F\) which can be applied to \(A\) (no force on \(B\) ) with no relative motion between \(A\) and \(B\) is \(0.84 \times 3 \times 9.8 \mathrm{~N}\) (d) both (b) and (c) are correct

Fine particles of a substance are to be stored in a heap on a horizontally circular plate of radius \(a\). if the coefficient of static friction between the particles is \(k\). The maximum possible height of cone is: (a) \(a k\) (b) \(\frac{a}{2} k\) (c) \(a / k\) (d) \(a k^{2}\)
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