Question

A uniform ladder stands on a rough floor and rests against a frictionless wall as shown in the figure. 2 58 1 Since the floor is rough, it exerts both a normal force N1 and a frictional force f1 on the ladder. However, since the wall is frictionless, it exerts only a normal force N2 on the ladder. The ladder has a length of L = 4.5 m, a weight of WL = 63.5 N, and rests against the wall a distance d = 3.75 m above the floor. If a person with a mass of m = 90 kg is standing on the ladder, determine the following.

(a) the forces exerted on the ladder when the person is halfway up the ladder (Enter the magnitude only.)

N1 = ____ N

N2 = ____ N

f1 = ____ N

(b) the forces exerted on the ladder when the person is three-fourths of the way up the ladder (Enter the magnitude only.)

N1 = ____ N

N2 = ____ N

f1 = ____ N

Answer 1

a)

N₁=Normal force from floor on ladder

N₂=Normal force from wall on ladder

f₁= Frictional force from floor on ladder

h=sqrt(L^2-d^2) = sqrt(4.5^2-3.75^2) = 2.5 m

By Newton’s 2^nd law along vertical,

F_nety = ma

N₁ – W_L – W_P = 0

N₁ = W_L + W_P

N₁ = W_L + mg

N₁ = 63.5 +90*9.8

N₁ = 945.5 N

Applying law of conservation of torque on at point of N₂

(W_L + W_P)*d/2- N₂*h = 0

N₂*h =(W_L + W_P)*d/2

N₂ = [(W_L + W_P)*d]/2h

Plugging values,

N₂ = [(63.5 +90*9.8)*3.75]/(2*2.5)

N₂ = 709.12 N

Applying Newton’s 2^nd law horizontally,

N₂ – f₁ = 0

f₁ = N₂

f₁ = 709.12 N

b)

N₁=Normal force from floor on ladder

N₂=Normal force from wall on ladder

f₁= Frictional force from floor on ladder

h=sqrt(L^2-d^2) = sqrt(4.5^2-3.75^2) = 2.5 m

By Newton’s 2^nd law along vertical,

F_nety = ma

N₁ – W_L – W_P = 0

N₁ = W_L + W_P

N₁ = W_L + mg

N₁ = 63.5 +90*9.8

N₁ = 945.5 N

Applying law of conservation of torque on at point of N₂

(W_L + W_P)*3d/4- N₂*h = 0

N₂*h =(W_L + W_P)*3d/4

N₂ = [(W_L + W_P)*3d]/4h

Plugging values,

N₂ = [(63.5 +90*9.8)*3*3.75]/(4*2.5)

N₂ = 1063.68 N

Applying Newton’s 2^nd law horizontally,

N₂ – f₁ = 0

f₁ = N₂

f₁ = 1063.68 N