A uniform ladder with mass m 2 and length L rests against a smooth wall. (Figure

Question

A uniform ladder with mass m2 and length L rests against a smooth wall. (Figure 1) A do-it-yourself enthusiast of mass m1 stands on the ladder a distance dfrom the bottom (measured along the ladder). The ladder makes an angle with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude f 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. Throughout the problem, consider counterclockwise torques to be positive. None of your answers should involve (i.e., simplify your trig functions).

Part A

What is the minimum coeffecient of static friction min required between the ladder and the ground so that the ladder does not slip?

Express min in terms of m1, m2, d, L, and .

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Part B

Suppose that the actual coefficent of friction is one and a half times as large as the value of min. That is, s=(3/2)min. Under these circumstances, what is the magnitude of the force of friction f that the floor applies to the ladder?

Express your answer in terms of m1, m2, d, L, g, and . Remember to pay attention to the relation of force and s.

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Figure 1 of 1

A uniform ladder with mass m2 and length L rests against a smooth wall. (Figure 1) A do-it-yourself enthusiast of mass m1 stands on the ladder a distance dfrom the bottom (measured along the ladder). The ladder makes an angle with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude f 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. Throughout the problem, consider counterclockwise torques to be positive. None of your answers should involve (i.e., simplify your trig functions).

Part A

What is the minimum coeffecient of static friction min required between the ladder and the ground so that the ladder does not slip?

Express min in terms of m1, m2, d, L, and .

min =

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Part B

Suppose that the actual coefficent of friction is one and a half times as large as the value of min. That is, s=(3/2)min. Under these circumstances, what is the magnitude of the force of friction f that the floor applies to the ladder?

Express your answer in terms of m1, m2, d, L, g, and . Remember to pay attention to the relation of force and s.

f =

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Figure 1 of 1

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Explanation / Answer

According to the given problem,

1)Balancing vertical forces,
N = m + m
When the ladder is on the verge of slipping, frictional force from right to left at the bottom of the ladder with the floor,
F = _min * N = _min * (m + m)
Taking moments about the point of contact of the ladder with the wall,
N * Lcos - F * Lsin - m * (L - d) cos - m * (L/2) cos = 0
=> (m+m) * Lcos - _min * (m+m) * Lsin - m * (L - d) cos - m * (L/2) cos = 0
=> _min * (m+m) * Lsin = (m+m) * Lcos - m * (L - d) cos - m * (L/2) cos
=> _min * (m+m) * Lsin = m * dcos + (m/2) Lcos

=> _min
= (dm + Lm/2)cot / [L(m+m)]
= [(d/L)m + (1/2)m] cot / (m+m).

2)Balancing vertical forces,
N = m + m
Balancing horizontal forces,
f = N
Taking moments of all the forces about the point of contact of the ladder with the floor,
- N Lsin + mg (L/2)cos + mg dcos = 0
=> f = N = (m/2 + md/L) gcot.

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