(10%) Problem 7: A uniform stationary ladder of length L and mass M leans agains

Question

(10%) Problem 7: A uniform stationary ladder of length L and mass M leans 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 . The ladder makes an angle with respect to the floor. A painter of weight 1/2M stands on the ladder a distance d from its base ©theexpertta.com × 25% Part (a) Find an expression for the magnitude of the normal force N exerted by the floor on the ladder Grade Summary Deductions Potential 5% 95% cos(a) sin() cos() sin() cos(0) tan(0) cotan(0) | ( Submissions Attempts remaining:4 % per attempt) detailed view 123 5% 0 SubmitHntFeedbackgive up! Hints: 1 % deduction per hint. Hi ints remaining: 4 Feedback: deduction per feedback. D 25% Part (b) Find an expression for the magnitude of the normal force NW exerted by the wall on the ladder. D 25% Part (c) Find an expression for the largest value of dmax for which the ladder does not slip - 25% Part (d) What is the largest value for d, in centimeters, such that the ladder will not slip? Assume that ladder is 3.5 m long, the coefficient of friction is 0.55, the ladder is at an angle of 46°, and the ladder has a mass of 55 kg

Explanation / Answer

Part A)

Normal force exerted by the floor on the ladder

N = (M + ½M)g

N = (3M*g)/ 2

Part B)

Sum the moments about the base of the ladder:

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)

Nw= [M*g*(1 + d/L)] / 2tan

Part C)

Nw = Ff

Ff = µ*N

Ff = µ*3Mg / 2

take the value of Nw from part B.

[M*g*(L+d)] / (2*L*tan) = µ*3*M*g / 2

(L+d) / (L*tan) = µ*3

L+d = 3*µ*L*tan

dmax = L*(3*µ*tan - 1)

Part D)

dmax = L*(3*µ*tan - 1)

dmax = 3.5m * (3*0.55*tan46º - 1)

dmax= 2.48 m

dmax= 248 cm

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