Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 05 - Using Newton's Laws: Friction, Circular Motion, Drag Forces

Problem 88

Chapter 5, Problem 88

The 70.0-kg climber in Fig. 5–53 is supported in the 'chimney' by the friction forces exerted on his shoes and back. The static coefficients of friction between his shoes and the wall, and between his back and the wall, are 0.80 and 0.60, respectively. What is the minimum normal force he must exert? Assume the walls are vertical and that the static friction forces are both at their maximum. Ignore his grip on the rope.

Verified step by step guidance

Identify the forces acting on the climber: The climber is in equilibrium, so the forces acting on him must balance. The forces include his weight (gravitational force), the frictional forces from his shoes and back, and the normal forces exerted by the walls on his shoes and back.

Write the equilibrium conditions: Since the climber is stationary, the net force in both the vertical and horizontal directions must be zero. For the vertical direction, the upward frictional forces must balance the downward gravitational force. For the horizontal direction, the normal forces exerted by the walls on his shoes and back must balance each other.

Express the frictional forces: The frictional force is given by $ f = \mu N $, where $ \mu $ is the coefficient of static friction and $ N $ is the normal force. For the shoes, $ f_{shoes} = \mu_{shoes} N_{shoes} $, and for the back, $ f_{back} = \mu_{back} N_{back} $.

Set up the vertical force balance: The total upward frictional force must equal the climber's weight. This gives $ \mu_{shoes} N_{shoes} + \mu_{back} N_{back} = mg $, where $ m $ is the climber's mass and $ g $ is the acceleration due to gravity.

Set up the horizontal force balance: The normal forces exerted by the walls on the climber's shoes and back must be equal in magnitude but opposite in direction. This gives $ N_{shoes} = N_{back} $. Substitute this relationship into the vertical force balance equation to solve for the minimum normal force $ N $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It acts parallel to the surfaces and is dependent on the normal force and the coefficient of static friction. The maximum static friction force can be calculated using the formula: F_friction = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force.

Normal Force

The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. In this scenario, the climber exerts a normal force against the walls, which is crucial for generating the static friction needed to prevent slipping. The magnitude of the normal force directly influences the maximum static friction that can be achieved.

Equilibrium Conditions

In physics, equilibrium refers to a state where the sum of forces and the sum of torques acting on an object are zero, resulting in no acceleration. For the climber, this means that the upward frictional forces must balance the downward gravitational force. Understanding equilibrium is essential to determine the minimum normal force required to maintain the climber's position without slipping.

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