Question

A 14.0 m uniform ladder weighing 480 N rests against a frictionless wall. The ladder makes a 63.0°-angle with the horizontal.

(a)

Find the horizontal and vertical forces (in N) the ground exerts on the base of the ladder when an 850-N firefighter has climbed 4.10 m along the ladder from the bottom.

horizontal force

magnitude Ndirection ---Select--- towards the wall away from the wall

vertical force

magnitude Ndirection ---Select--- up down

(b)

If the ladder is just on the verge of slipping when the firefighter is 9.20 m from the bottom, what is the coefficient of static friction between ladder and ground?

(c)

What If? If oil is spilled on the ground, causing the coefficient of static friction to drop to half the value found in part (b), what is the maximum distance (in m) the firefighter can climb along the ladder from the bottom before the ladder slips?

m

Answer 1

Given,

Weight of the lader, W_L = 480 N

Length of the ladder, L = 14 m

= 63

a)

Weight of the man, W_M = 850 N

Thus,

Normal force, N = Weight of the man + weight of the ladder

                          = 850 + 480 = 1330 N

Vertical force is up

There is a frictional force in the direction towards the wall

Let coefficient of static friction be

Frictional force, F_fr = *N

                               = ( * 1330) N

b)

Distance firefighter is from bottom, d = 9.20 m

AS we know, ladder is stationary, hence

N_W = *N = *1330

Taking point A as the pivot

Since, ladder is stationary,

Total torque = 0

=> (Weight of the ladder * cos)* L/2 + (Weight of the man * cos)* d - ( N_W * sin)* L = 0

=> 480*cos63 * (14/2) + 850*cos63*9.20 - ( N_W *sin63 * 14) = 0

=> N_W *0.891 * 14 = 480*0.454 * 7 + 850*0.454*9.20

=> 12.474 * N_W = 1525.44 + 3550.28 = 5075.72

=> N_W = 5075.72 / 12.474 = 406.904

=> *1330 = 406.904

=> = 406.904 / 1330 = 0.306

Thus, coefficient of static friction is 0.306

c)

Let the maximim height be d_max

Coefficient of friction is halfed, thus

₁ = /2 = 0.306 / 2 = 0.153

Now,

N_W = ₁*N = 1330 * 0.153 = 203.49

=> (Weight of the ladder*cos)* L/2 + (Weight of the man*cos)* d_max - ( N_W*sin)* L = 0

=> 480*cos63 * (14/2) + 850*cos63*d_max - ( 203.49 *sin63 * 14) = 0

=> 480*0.454*7 + 850*0.454*d_max = 203.49 *0.891*14

=> 1525.44 + 385.9*d_max = 2538.33

=> 385.9*d_max = 2538.33 - 1525.44 = 1012.89

=> d_max = 1012.89 / 385.9 = 2.623 m

or maximum distance is 2.623 m