Please help solving All parts . Thank you (10%) Problem 7: A uniform stationary

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Please help solving All parts .

Thank you

(10%) Problem 7: A uniform stationary ladder of length Land mass Mlean against a smooth vertical wall, while its bottom legs rest on a rough horizontal floor The coefficient of static friction between floor and ladder is u The ladder makes an angle with respect to the floor. A painter of weight 1/2Mstands on the ladder a distance d from its base Otheexpertta.co A 25% Part (a) Find an expression for the magnitude of the normal force N exerted by the floor on the ladder Grade Summary Deductions 100% Potential cos(a) cos(p) cos(0) cotana(0) Submissions Attempts remaining: 5 sin (p) sin(e) tan(0) 4 5 6 6% per attempt) 1 2 3 detailed view Submit Hint give up! Hints: 0% deduction per hint. Hints remaining: 4 Feedback: 0% deduction per feedback. A 25% Part b) Find an expression for the magnitude of the nomal force NiT exerted by the wall on the ladder. A 25% Part (c) Find an expression for the largest value of d for which the ladder does not slip. max A 25% Part (d) What is the largest value for d. in centimeters, such that the ladder will not slip? Assume that ladder is 4.5 m long, the coefficient of friction is 0.51 the ladder is at an angle of 49. 5 and the ladder has a mass of 55 kg.

Explanation / Answer

  "smooth vertical wall" means there is no friction there, so the only vertical forces are the weights of the ladder and painter and the normal force at the floor.

"Find an expression for the magnitude of the normal force, N, exerted by the floor on the ladder."
As stated above, we know that
N = (M + ½M)g = 3Mg / 2

"Find an expression for the magnitude of the normal force, Nw, exerted by the wall on the ladder."
Sum the moments about the base of the ladder:
M = 0 = M*g*L/2*cos + ½M*g*d*cos - Nw*L*sin
0 = ½M*g*cos*(L + d) - Nw*L*sin
Nw = M*g*(L+d) / 2*L*tan = M*g*(1 + d/L) / 2tan

"Find an expression of the largest value of d-max for which the ladder does not slip."
Since they are the only two horizontal forces in play, we know that
Nw = Ff where Ff is the friction force at the floor, and Ff = µ*N = µ*3Mg / 2. So
M*g*(L+d) / 2*L*tan = µ*3*M*g / 2 M*g/2 cancels
(L+d) / L*tan = µ*3
L+d = 3µLtan
max d = L(3µtan - 1)

"What is the largest value for d, in cm, such that the ladder will not slip?"
max d = 4.5m * (3*0.51*tan49.5º - 1) =3.5 m =350 cm

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