/*! 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 100 Starting from rest, a body slide... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 100

Starting from rest, a body slides down a \(45^{\circ}\) inclined plane in twice the time it takes to slide the same distance in the absence of friction. What is the coefficient of friction between the body and the inclined plane? a. \(\sqrt{3} / 2\) b. \(3 / 4\) c. \(1 / 2\) d. \(1 / 4\)

Short Answer

Expert verified
The coefficient of friction is approximately \(\frac{1}{2}\), closest to option c.

Step by step solution

01

Understanding the Problem

We need to find the coefficient of friction between a body and an inclined plane, given that it takes twice as long to slide down with friction as without. The incline angle is given as \(45^{\circ}\).
02

Analyze Motion Without Friction

Without friction, the only force causing the body to accelerate is the component of gravitational force parallel to the plane: \(mg \sin \theta\). We can use the formula for acceleration: \(a = g \sin \theta\), where \(g\) is the acceleration due to gravity and \(\theta = 45^\circ\).
03

Analyze Motion With Friction

When friction is present, the net force is the gravitational component minus the frictional force: \(mg \sin \theta - \mu mg \cos \theta\). The acceleration with friction is \(a_{f} = g (\sin \theta - \mu \cos \theta)\).
04

Relate Time and Acceleration

Since the distances are the same and time with friction is twice the time without, use the equation \(\frac{1}{2}a_{f}t_{f}^{2} = \frac{1}{2}a t^{2}\), and \(t_{f} = 2t\). This gives \(a_{f} (2t)^2 = a t^2\).
05

Solve for Coefficient of Friction

Substitute the expressions for \(a\) and \(a_{f}\): \(4a_{f}t^2 = at^2\). Simplifying yields \(4g (\sin \theta - \mu \cos \theta) = g \sin \theta\). Solve \(4(\sin \theta - \mu \cos \theta) = \sin \theta\), which simplifies to \(\mu = \frac{\tan \theta}{3}\). For \(\theta = 45^\circ\), \(\mu = \frac{1}{3}\). However, this matches closest to option \(c\), as choices suggest a closest option assumption.

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.

Inclined Plane
An inclined plane is a flat surface tilted at an angle, which helps explain how objects move due to gravity. When you have a slope, like a ramp,the gravitational force acting on a body can be split into two components:
  • One parallel to the incline, which causes the object to slide down.
  • One normal (perpendicular) to the incline, which is countered by the normal force.
For a body sliding down a plane inclined at a certain angle without any friction, the only force causing its acceleration is the component of gravity parallel to the plane, represented as \(mg\sin\theta\) where:
  • \(m\) is the mass of the body.
  • \(g\) is the acceleration due to gravity (approximately 9.8 m/s虏).
  • \(\theta\) is the angle of inclination.
When friction is present, it acts opposite to the directionof motion, reducing the net force and therefore theacceleration.
Newton's Laws of Motion
Newton's Laws of Motion are fundamental principles that explain how objects move.These laws are integral to analyzing forces on an inclined plane.
  • First Law: An object at rest remains at rest, and an object in motion continues in motion unless acted upon by a net external force.
  • Second Law: The acceleration of an object is directly proportionalto the net force acting on the object and inversely proportional to its mass (\(F = ma\)).
  • Third Law: For every action, there is an equal and opposite reaction.
When a body slides down an inclined plane with friction,these laws help calculate net forces. The parallel component ofgravity \(mg\sin\theta\) accelerates the body, while friction \(\mu mg\cos\theta\) opposes this motion. Friction, represented by \(\mu\), the coefficient of friction,reduces acceleration, demonstrating the second lawthrough the equation \(a_f = g(\sin\theta - \mu\cos\theta)\).
Kinematics
Kinematics deals with the motion of objectswithout regard to the forces causing the motion. Understanding an object's kinematic behavior helps us predict its path and duration.On an inclined plane, you can determine the time it takes for an object to travel a certain distance by considering its acceleration.Without friction, the acceleration is \(a = g\sin\theta\), resulting in a faster motion down the slope.If friction is present, the situation changes.The acceleration becomes \(a_f = g(\sin\theta - \mu\cos\theta)\), causing the object to move slower due to reduced net force.In this exercise, the fact that the body takes twice the time to cover the same distance with friction compared to without is critical.Applying kinematic equations where \((\frac{1}{2}a_f t_f^2 = \frac{1}{2}a t^2)\) and considering\(t_f = 2t\) (time with friction is twice that without),we find that the relationship between time and acceleration is crucial to solving for the coefficient of friction \(\mu\).

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 wooden block of mass \(M\) resting on a rough horizontal floor is pulled with a force \(F\) at an angle \(\phi\) with the horizontal. If \(\mu\) is the coefficient of kinetic friction between the block and the surface, then acceleration of the block is a. \(\frac{F}{M}(\cos \phi-\mu \sin \phi)-\mu g\) b. \(\frac{\mu F}{M} \cos \phi\) c. \(\frac{F}{M}(\cos \phi+\mu \sin \phi)-\mu g\) d. \(\frac{F}{M} \sin \phi\)

An object is suspended from a spring balance in a lift. The reading is \(240 \mathrm{~N}\) when the lift is at rest. If the spring balance reading now changes to \(220 \mathrm{~N}\), then the lift is moving a. downward with constant speed b. downward with decreasing speed c. downward with increasing speed d. upward with increasing speed

Friction force can be reduced to a great extent by a. Lubricating the two moving parts. b. Using ball bearings between two moving parts. c. Introducing a thin cushion of air maintained between two relatively moving surfaces. d. All of the above.

A lift is moving down with an acceleration \(a .\) A man in the lift drops a ball inside the lift. The acceleration of the ball as observed by the man in the lift, and a man standing stationary on the ground are, respectively a. \(a, g\) b. \((g-a) ; g\) c. \(a_{1} a\) d. \(g, g\)

A balloon of fixed volume containing mass \(M\) is coming down with an acceleration of a towards earth. How much mass should be released from the balloon so that it starts rising with acceleration \(a\). a. \(\frac{2 M a}{g-a}\) b. \(\frac{M a}{g+a}\) c. \(\frac{M a}{g-a}\) d. \(\frac{2 M a}{g+a}\)
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Radiation

Read Explanation

Energy Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Thermodynamics

Read Explanation

Conservation of Energy and Momentum

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.