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A-uniform-ladder-of-weight-W-and-length-2a-rest-in-limiting-equilibrium-with-one-end-on-a-rough-horizontal-ground-and-the-other-end-on-a-rough-vertical-wall-The-coefficient-of-friction-between-the-l




Question Number 76579 by Rio Michael last updated on 28/Dec/19
A uniform ladder of weight W and length  2a rest in limiting equilibrium with one   end on a rough horizontal ground and the  other end on a rough vertical wall.  The coefficient of friction between the ladder  and the ground and between the ladder and  the wall are respectively μ and λ . If the ladder  makes an angle θ with the ground where tan θ = (5/(12)),  a) show that 5μ + 6λμ − 6 = 0.  b) find the value of λ and μ given that λμ =(1/2).
AuniformladderofweightWandlength2arestinlimitingequilibriumwithoneendonaroughhorizontalgroundandtheotherendonaroughverticalwall.Thecoefficientoffrictionbetweentheladderandthegroundandbetweentheladderandthewallarerespectivelyμandλ.Iftheladdermakesanangleθwiththegroundwheretanθ=512,a)showthat5μ+6λμ6=0.b)findthevalueofλandμgiventhatλμ=12.
Answered by mr W last updated on 28/Dec/19
Commented by mr W last updated on 28/Dec/19
f_1 =μN_1   f_2 =λN_2   N_1 +f_2 =W ⇒N_1 +λN_2 =W   ...(i)  N_2 −f_1 =0 ⇒ N_2 −μN_1 =0   ...(ii)  L=length of ladder  W×(L/2) cos θ+f_1 ×L sin θ−N_1 ×L cos θ=0  W cos θ+2 (μ sin θ−cos θ)N_1 =0   ...(iii)  (i)−(ii)×λ:  N_1 (1+μλ)=W  put this into (iii):  N_1 (1+μλ) cos θ+2 (μ sin θ−cos θ)N_1 =0  (1+μλ) cos θ+2 (μ sin θ−cos θ)=0  2μ sin θ=(1−μλ) cos θ  ⇒tan θ=((1−μλ)/(2μ))=(5/(12))  ⇒5μ+6μλ−6=0    if μλ=(1/2)  5μ+6×(1/2)−6=0  ⇒μ=(3/5)  ⇒λ=(1/(2μ))=(5/6)
f1=μN1f2=λN2N1+f2=WN1+λN2=W(i)N2f1=0N2μN1=0(ii)L=lengthofladderW×L2cosθ+f1×LsinθN1×Lcosθ=0Wcosθ+2(μsinθcosθ)N1=0(iii)(i)(ii)×λ:N1(1+μλ)=Wputthisinto(iii):N1(1+μλ)cosθ+2(μsinθcosθ)N1=0(1+μλ)cosθ+2(μsinθcosθ)=02μsinθ=(1μλ)cosθtanθ=1μλ2μ=5125μ+6μλ6=0ifμλ=125μ+6×126=0μ=35λ=12μ=56
Commented by Rio Michael last updated on 28/Dec/19
thanks sir
thankssir

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