/*! 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 71 A uniform ladder of length \(L\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 71

A uniform ladder of length \(L\) and weight \(w\) is leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is the same as that between the ladder and the wall. If this coefficient of static friction is \(\mu_{s}=0.500\), determine the smallest angle the ladder can make with the floor without slipping.

Short Answer

Expert verified
The smallest angle the ladder can make with the floor without slipping is found by equating the forces and torques acting on the ladder, which gives us the relation \(\theta = \arctan(2\mu_s)\). With the given value of \(\mu_s = 0.500\), this yields \( \theta = 63.4^{\circ} \).

Step by step solution

01

Label and draw a diagram

Draw a diagram of the ladder leaning against the wall. Label this with the forces acting on it. For the ladder: The weight \(w\) acts downwards from its center, the normal force \(N_1\) acts upwards at the contact point with the floor, the frictional force \(\mu_{s}*N_1\) to the right from the same point. At the point where the ladder touches the wall: the normal force \(N_2\) acts to the left, and the frictional force \(\mu_{s}*N_2\) acts upwards.
02

Setup the equations of equilibrium

We can write two conditions for equilibrium. The net force and net torque on the ladder must be zero. Forces: Net force in the x-direction: \(\mu_{s}*N_1 - N_2 = 0\) (eq.1); net force in the y direction: \(N_1 - w + \mu_{s}*N_2= 0\) (eq.2). Torques: Taking torques about contact point with ground: \( \mu_{s}*N_2* L/2 \sin(\theta) - N_2 * L/2 \cos(\theta) - w * L/2 = 0\) (eq.3). Here, \(\theta\) is the angle the ladder makes with the ground.
03

Solve the equations

Extract \(N_1\) from eq.1 and \(N_2\) from eq.2 and finally substitute these into eq.3. With \(\mu_{s}=0.500\) and \(w\) cancelling out, we can solve for \(\theta\) which will give us the smallest angle the ladder can make with the floor without slipping.

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 Static Friction
Understanding the coefficient of static friction is crucial when analyzing scenarios like a ladder leaning against a wall. The coefficient of static friction, denoted as \(\mu_s\), is a dimensionless value representing the ratio between the maximum static frictional force and the normal force. In simpler terms, it quantifies the

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 \(12.0\) -kg object is attached to a cord that is wrapped around a wheel of radius \(r=10.0 \mathrm{~cm}\) (Fig. P8.70). The acceleration of the object down the frictionless incline is measured to be \(2.00 \mathrm{~m} / \mathrm{s}^{2}\). Assuming the axle of the wheel to be frictionless, determine (a) the tension in the rope, (b) the moment of inertia of the wheel, and (c) the angular speed of the wheel \(2.00 \mathrm{~s}\) after it begins rotating, starting from rest.

The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. \(\mathrm{P} 8.29 \mathrm{a}\) ). The forces on the lower leg when the leg is extended are modeled as in Figure P8.29b, where \(\vec{T}\) is the force of tension in the tendon, \(\vec{w}\) is the force of gravity acting on the lower leg, and \(\overrightarrow{\mathbf{F}}\) is the force of gravity acting on the foot. Find \(\vec{T}\) when the tendon is at an angle of \(25.0^{\circ}\) with the tibia, assuming that \(w=30.0 \mathrm{~N}\), \(F=12.5 \mathrm{~N}\), and the leg is extended at an angle \(\theta\) of \(40.0^{\circ}\) with the vertical. Assume that the center of gravity of the lower leg is at its center and that the tendon attaches to the lower leg at a point one-fifth of the way down the leg

A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs \(200 \mathrm{~N}\) and is \(3.00 \mathrm{~m}\) long. What is the tension in each rope when the \(700-\mathrm{N}\) worker stands \(1.00 \mathrm{~m}\) from one end?

A merry-go-round rotates at the rate of \(0.20 \mathrm{rev} / \mathrm{s}\) with an \(80-\mathrm{kg}\) man standing at a point \(2.0 \mathrm{~m}\) from the axis of rotation. (a) What is the new angular speed when the man walks to a point \(1.0 \mathrm{~m}\) from the center? Assume that the merry-go-round is a solid \(25-\mathrm{kg}\) cylinder of radius \(2.0 \mathrm{~m} .\) (b) Calculate the change in kinetic energy due to the man's movement. How do you account for this change in kinetic energy?

A hungry \(700-\mathrm{N}\) bear walks out on a beam in an attempt to retrieve some "goodies" hanging at the end (Fig. P8.22). The beam is uniform, weighs \(200 \mathrm{~N}\), and is \(6.00 \mathrm{~m}\) long; the goodies weigh \(80.0 \mathrm{~N}\). (a) Draw a free-body diagram of the beam. (b) When the bear is at \(x=1.00 \mathrm{~m}\), find the tension in the wire and the components of the reaction force at the hinge. (c) If the wire can withstand a maximum tension of \(900 \mathrm{~N}\), what is the maximum distance the bear can walk before the wire breaks?
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Modern Physics

Read Explanation

Fields in Physics

Read Explanation

Physics of Motion

Read Explanation

Solid State Physics

Read Explanation

Rotational 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.