(8%) Problem 7: A uniform stationary ladder of length L 3.4 m and mass M-19 kg l
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Question
(8%) Problem 7: A uniform stationary ladder of length L 3.4 m and mass M-19 kg 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 u- a39. The ladder makes an angle ?-51° with respect to the floor. A painter of mass 8M stands on the ladder a distance d from its base Otheexpertta.com 33% Part (a) Find the magnitude of the normal force N, in newtons, exerted by the floor on the ladder. Grade Summary 090 100% Potential sin0 cotan asinacos atan acotan sinh0 cosh t cotanh Submissions Attempts remaining:7 0% per attempt) coso tan ( 7 89 detailed view 0 END Degrees Radians VO DEL? CL.EAR Submit Hint I give up Hints: .0% deduction per hint. Hints remaining: 2 Feedback: 2% deduction per feedback. 3300 Part (b) Find an expression for the magnitude of the normal force MW exerted by the wall on the ladder ? 33% Part (c) what is the largest distance up the ladder d.mar, in meters, that the painter can stand without the ladder slipping?Explanation / Answer
a) Since the wall is frictionless, the normal force at the floor is the sum of the weights of the ladder and painter:
Fn = (1 + 8) * 19kg * 9.8m/s² = 1675.8 N ?
b)
Sum the moments about the base of the ladder:
?M = 0 = M*g*L/2*cos? + 8M*g*d*cos? - Nw*L*sin?
0 = M*g*cos?*(L/2 + 8d) - Nw*L*sin?
Nw = M*g*(L/2 + 8d) / L*tan?
= M*g*(1/2 + 8d/L) / tan?
c) For horizontal equilibrium, the normal force at the wall (Fw) must equal the friction force at the floor; at the threshold of slipping, that's
Fw = f = µ*Fn = 0.39 * 1675.8N = 653.56 N
Now sum the moments about the base of the ladder:
?M = 0 = 653.56N*3.4m*sin51º - (19kg*3.4m/2 + 8*19kg*d)*9.8m/s²*cos51º
This solves to
d = 1.63 m ?
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