Question

You are in your room when the fire alarm sounds. You look out of your door and see the stairs full of smoke. You remember that some work was being done on the outside of the building and a 10 m long ladder is reachable outside your window. You can climb out to a point 2 m from the top of the ladder. you see that the ladder is resting against a smooth wall (negligible friction) above you and on a rough surface on the ground. You guess that the coefficient of static friction between the ladder and the ground is 0.5. You weigh 80 kg and you see written on the side of the ladder that it weighs 10 kg. The ladder is set up at an angle of 45 degrees to the ground. You are afraid that if you get on the ladder it will slide from under you, but you see a friend with a rope that you had recently tested in the lab to have a breaking tension of 300 N and you shout to her to tie the bottom of the ladder to a peg at the base of the wall. You calculate that this should hold and stop the ladder sliding. You climb out onto the ladder.

a. Would the ladder have fallen without the rope?

b. will the rope break?

Answer 1

Forces acting on the ladder are shown in the figure, with assumption that ladder is in euqilibrium.
Force from ground in Vertical direction N = 90g, balancing the weight of Man and ladder.
Force fro ground in horizontal direction, that is friction f is same as force by wall on ladder in horizontal direction, as net force on the ladder in horizontal direction is zero.

With N= 90g and and coefficient of static friction (mew) = 0.5, maximum force of frction that can act on ladder is
(mew) N = 45g = 441 N.

If ladder has to stay in equilibrium without rope tied at bottom, Force of friction, with ladder in equilibrium must be less than 441 N. To find force of friction, we write net torque about point 'A' at equate it to zero.
Torque is force x (perpendicular distance of force from reference). Taking clockwise torque as +ve and anticlockwise as -ve we get

10g (5/2^1/2 ) + 80g(8/2^1/2) - f (10/2^1/2) = 0
f = 676.2 N
As this is more than maximum friction, Ladder will fall without rope.

b) Wth rope in position, maximum force that can be applied by friction + rope , becomes 441 + 300 = 771N.
But required force at the bottom of Ladder, to keep it in equilibrium remains same same, that is 676.2. So Ladder will remain iin equilibrium and rope will be under tension of 676.2 - 441 = 235.2 N