1) A ladder of mass m and length L leans against a frictionless wall and makes a

Question

1) A ladder of mass m and length L leans against a frictionless wall and makes an angle of 45 degrees with it. Determine the minimum coefficient of static friction with the floor necessary to prevent it from sliding.

2) In the previous problem, suppose a person of mass M is on the ladder. How far up the ladder can they stand before it slips?

Please show and explain the free body diagrams because im not sure how to set them up, along with an explanation of how to attempt/solve the problem. I know I have to uses the sum of all the forces and the sum of all the torques somehow and that the total force and torques are equal to 0

Explanation / Answer

1)
let F is the force exerted by wall on the ladder

as the ladder is in equilibrium net torque abouut any point is equal to zero.


m*g*(L/2)*sin(45) - F*L*sin(45) = 0


m*g*(L/2)*sin(45) = F*L*sin(45)


==> F = m*g/2

in order to keep ladder to be in equilbrium ftrictional force should be os the same as F

F = mue*m*g

m*g/2 = mue*m*g

mue = 0.5

2)

let the person can reach x distance on the ladder

let F is the force exerted by wall on the ladder

as the ladder is in equilibrium net torque abouut any point is equal to zero.


m*g*(L/2)*sin(45) + M*g*x*sin(45)- F*L*sin(45) = 0


m*g*(L/2)*sin(45) + M*g*x*sin(45)- mue*m*g*L*sin(45) = 0


m*g*(L/2)*sin(45) + M*g*x*sin(45)- 0.5*m*g*L*sin(45) = 0


solving the above equation we get x = 0.

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