/*! 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

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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.

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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

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Most popular questions from this chapter

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