# Homework Help: Ladder Problem using Torque

1. Dec 3, 2012

### dolpho

1. The problem statement, all variables and given/known data

A 5 meter long ladder, weighting 200 N rests against a smooth vertical wall with its base on a horizontal rough floor, a distance of 1.2 meters away from the wall. The center of mass of the ladder is 2.5 m from it's base, and the coefficient of friction is .20. How far up the ladder can a 600 N person climb before the ladder begins to tip.

2. Relevant equations
Ffriction = uN
F=ma
torque = RF

3. The attempt at a solution

So first I try identifying all of the forces acting on the ladder. In the X direction we have the Frictional force, and the wall acting on the ladder. In the vertical direction we have 200N from the ladder, the 600 N person, and the Normal force acting up on the base of the ladder.

The angle it makes with the wall is 76.1 degrees.

I use newtons Second law for each of these;

Fy = Ffriction - Fwall = 0
Ffriction = Fwall

Fx = N - 200 - 600 = 0
N = 800 Newtons

Now hopefully I did that part right but when I sum the torques I get confused. So I guess I need to set an axis and then pick which directions are positive or not. I pick the Normal force as my axis. I'll pick counter clockwise as positive and clockwise negative.

Sum of Torques = N(0) - 200 (2.5) - 600d + Fwall(5) = 0

The Fwall is equal to uN = (.2)(800)(cos 76)??

Yea I'm pretty lost here. Any help would be appreciated!

2. Dec 3, 2012

### PhanthomJay

yes
yes
and yet another yes!
well, his first, which is not unlike his 2nd when a = 0.
yes, but you have reversed x and y
Moment is force times perpendicular distance to the pivot...try that one again.
why no, the friction force is μN, and that is a horizontal force at the base of the ladder, and equal to Fwall...why the cos term?
I think you'll be unlost in a bit, i hope...

3. Dec 3, 2012

### dolpho

Yea sorry I typed it out wrong. So in the Y direction we have----> N - 200 - 800 = 0
So N=800

in the X we have Ffriction = Fwall

Now for the summing torques. Since the two forces pushing in the clockwise direction, would they have the same angle?

I redid the perpindicular force and hopefully found that its cos(13.9) = 800 / Fperp = 824 n ?

So for the ladder it would be -- > cos(13.9) = 200 / Fperp = 206 n ?

Sum of Torques = -824d - (206)(2.5) + Fwall (5) = 0

The torque of the Fwall would be umm....Fwall / cos13.9 ?

And the Fwall would be equal to the Frictional force since it isn't moving right? So F friction = (.2)(200 Newtons) = 40?

So I guess the total sum of the torques would be...

Sum Torque = -824d - 516 N + 1026N = 0

Dangit I know that's wrong. lol :(

(edit: Do I have all of the correct torques on the equation? So the normal force and Ffrictional force go away since I set is as my axis. So that just leaves the persons torque, the ladders torque and the torque of the wall ?)

Last edited: Dec 3, 2012
4. Dec 3, 2012

### PhanthomJay

At the base of the ladder, you already calculated the vertical reaction to be 800 Newtons. That vertical reaction IS the Normal force at that point. The normal force acts perpendicular to the contact surface (the floor), not along the slope of the ladder.
no....
no...ooo
Yes this part is more or less correct, when summing torques about the base of the ladder, the vertical normal force and the horizontal friction force of the floor on the ladder have no torque about the base, since the loads pass through that point.

There are 2 issues here: solving for the horizontal and vertical reaction forces of the floor on the ladder at the base of the ladder, and solving for the horizontal reaction force of the wall on the ladder. The vertical normal force is __??___ N and then the limiting friction force is___??___, in which direction? Then once you know the friction force, then the wall force must be equal to it and in which direction?

Then now you should sum torques of the ladder weight, the man's weight, and the wall force, about the ladder base. There are 2 ways to calculate torques. The first is to use the cross product rule T = r X F = rFsintheta, where r is the magnitude of the position vector distance between the point of force application and the point you are summing torques about, and theta is the included angle between the force and position vector. The second way is to use T = force times perpendicular distance from the line of action of the force to the point about which you are summing torques. Let me start you off by calculating the torque about the base of the ladder's weight: Using the first method, T = r X F = rF sintheta = (2.5)(200) sin 13.9. Using the 2nd method, T = F*(perp. distance) = 200(2.5)cos 76.1. Both yield the same result. Watch cw vs. ccw plus or minus signs. It may be a bit easier here to use the first method for the calcs, although the 2nd gives a better understanding of the concept, but with some added geometry/trig. Continue...

5. Dec 4, 2012

### dolpho

Ok I think I got this now...

Sum Torques = Fwall(5) - (200N)(2.5) - 600(x) = 0

And then I find the perpendicular forces of the above which is...

Fwall = Sin 76.1 x 800(.2) = 155 N
Ladder force = Sin13.9 x 200(2.5)
Person force = Sin13.9 x 600(d)

Sum Torque = 775 - 120 - 144d = 0
d=4.55

Hopefully that's right but one question I had was when I was trying to find the perpindicular force for the first one, I had a right triangle and I did...

Sin(76.1) = Fperp / 160 = 155

But if I took the cosine wouldn't it be the same answer?

Cos(13.9) = Fwall / 160 = 165?

Sorry if it's a stupid question I always thought that it didn't matter if you took sine or cosine. Thanks for the help!

6. Dec 4, 2012

### PhanthomJay

scratch this equation please it is not correct
well I gave you two ways to find torque, and darned you found a third way by finding the perpendicular component of the force to the ladder and using the distance along the ladder, which is ok to do also since the component of the force parallel to the ladder produces no torque.
yes, good work
check your calculator again that comes out to 155 same result
no such thing as a stupid question. Sin theta = cos(90 - theta), you just slipped a digit when using your calculator.

7. Dec 4, 2012