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A uniform ladder with mass m2 = 14kg and length L = 3.2m rests against a smooth wall. A do-it-yourself enthusiast of mass m₁ = 70kg stands on the ladder a distance d = 1.2m from the bottom (measured along the ladder.) There is no friction between the wall and the ladder, but there is a frictional force, with µ = 0.25, between the floor and the ladder. N₁ is the magnitude of the normal force exerted by the wall on the ladder, and N₂ is the magnitude of the normal force exerted by the ground on the ladder. What angle 6, will cause the static friction with the floor to be maximized? Will the ladder start to slip when the angle is made larger, or smaller than this borderline value?

A uniform ladder with mass m2 = 14kg and length L = 3.2m rests against a smooth wall. A do-it-yourself enthusiast of mass m₁ = 70kg stands on the ladder a distance d = 1.2m from the bottom (measured along the ladder.) There is no friction between the wall and the ladder, but there is a frictional force, with µ = 0.25, between the floor and the ladder. N₁ is the magnitude of the normal force exerted by the wall on the ladder, and N₂ is the magnitude of the normal force exerted by the ground on the ladder. What angle 6, will cause the static friction with the floor to be maximized? Will the ladder start to slip when the angle is made larger, or smaller than this borderline value?

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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A uniform ladder with mass \( m_2 = 14 \, \text{kg} \) and length \( L = 3.2 \, \text{m} \) rests against a smooth wall. A do-it-yourself enthusiast of mass \( m_1 = 70 \, \text{kg} \) stands on the ladder a distance \( d = 1.2 \, \text{m} \) from the bottom (measured along the ladder). There is no friction between the wall and the ladder, but there is a frictional force, with \( \mu = 0.25 \), between the floor and the ladder. \( N_1 \) is the magnitude of the normal force exerted by the wall on the ladder, and \( N_2 \) is the magnitude of the normal force exerted by the ground on the ladder. What angle \( \theta \), will cause the static friction with the floor to be maximized? Will the ladder start to slip when the angle is made larger, or smaller than this borderline value? **Diagram Explanation:** The diagram shows a ladder leaning against a wall, with several forces acting on it: - \( \vec{N_1} \): Normal force exerted by the wall, acting horizontally on the ladder. - \( \vec{N_2} \): Normal force exerted by the floor, acting vertically on the ladder. - \( \vec{f} \): Frictional force at the base of the ladder, acting horizontally. - \( \vec{m_1g} \): Gravitational force acting downward at the point where the person stands. - \( \vec{m_2g} \): Gravitational force acting downward at the ladder's center of mass. **Instructions:** 1) Draw a free body diagram for the ladder, choose a coordinate system and write out the equations for the sum of the x forces and the sum of the y forces. 2) Draw a diagram of the ladder with each force vector acting on it at its point of application. Choose an axis of rotation and place this on the diagram as well. (Note: Any choice will work, you just have to be consistent later.) 3) Find the length of the lever arm for each force acting on the ladder and also list whether that force would tend to cause clockwise or counter-clockwise rotation. 4) Write out the net torque equation
Transcribed Image Text:A uniform ladder with mass \( m_2 = 14 \, \text{kg} \) and length \( L = 3.2 \, \text{m} \) rests against a smooth wall. A do-it-yourself enthusiast of mass \( m_1 = 70 \, \text{kg} \) stands on the ladder a distance \( d = 1.2 \, \text{m} \) from the bottom (measured along the ladder). There is no friction between the wall and the ladder, but there is a frictional force, with \( \mu = 0.25 \), between the floor and the ladder. \( N_1 \) is the magnitude of the normal force exerted by the wall on the ladder, and \( N_2 \) is the magnitude of the normal force exerted by the ground on the ladder. What angle \( \theta \), will cause the static friction with the floor to be maximized? Will the ladder start to slip when the angle is made larger, or smaller than this borderline value? **Diagram Explanation:** The diagram shows a ladder leaning against a wall, with several forces acting on it: - \( \vec{N_1} \): Normal force exerted by the wall, acting horizontally on the ladder. - \( \vec{N_2} \): Normal force exerted by the floor, acting vertically on the ladder. - \( \vec{f} \): Frictional force at the base of the ladder, acting horizontally. - \( \vec{m_1g} \): Gravitational force acting downward at the point where the person stands. - \( \vec{m_2g} \): Gravitational force acting downward at the ladder's center of mass. **Instructions:** 1) Draw a free body diagram for the ladder, choose a coordinate system and write out the equations for the sum of the x forces and the sum of the y forces. 2) Draw a diagram of the ladder with each force vector acting on it at its point of application. Choose an axis of rotation and place this on the diagram as well. (Note: Any choice will work, you just have to be consistent later.) 3) Find the length of the lever arm for each force acting on the ladder and also list whether that force would tend to cause clockwise or counter-clockwise rotation. 4) Write out the net torque equation
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