Where Should a 150N Weight Be Placed on a Ladder to Achieve Equilibrium?

[learn2physics]

Homework Statement

A ladder weighing 200N is being held by a man. The man shoulder acts as a pivot. His arm is pushing an upward force F.

http://i.imgur.com/dMrcRtK.jpg

Where could a weight of 150N be hang upon, so that force F do not need to counteract the weight of the ladder

Homework Equations

Moment= F x Perpendicular Distance

The Attempt at a Solution

I am not sure if I am understanding it correctly. If I am not wrong, the question is asking me, where can I hang a weight of 150N, so that force F no need to counteract the weight of the ladder. Meaning, assume force F = 0N.

Let the distance be x
Total clockwise moment = total anti clockwise moment
150x = 200*0.5
150x = 100
x = 2/3
= 0.667m.

Therefore it needs to be hang 0.667m away from the pivot or at 3.167m of the ladder.

Am I understanding this correctly?

 

[Nathanael]

From your picture, I thought the pivot was at 0.25m (that's where "F" is drawn) but I guess that 0.25m is just the length of his arm?

Anyway, if the pivot is at 0.5m then I believe you've done it correctly, because I did it a different way and got the same answer. (I could have made a mistake, but as far as I can tell it is correct.)

 

[learn2physics]

Yes the pivot is at 2.5 mark of the ladder from the left. Sorry my picture wasn't clear.

Is my understanding correct? Thank you!

 

[Nathanael]

It appears to be, from your math. But there are usually multiple ways to understand something.

For example, you used the center of mass of the entire ladder to solve the problem, whereas I solved it using the center of mass of only the first meter (from the left) of the ladder.

The reason is, in my eyes, the first meter (from the left) is the only important part (because the other 3 meters are symmetrical around the pivot and therefore cancel out). So I took the mass of the first meter (50 Newtons, since it's 1/4 of the entire ladder) and multiplied it by the distance from the pivot (2 meters) and then I set it equal to 150xWe created our equations based on different understandings, yet they still led to the same answer.

 

[HalfLostAndSearching]

Yes, you are correct in your understanding of the question and your solution. The key concept here is the principle of moments, which states that for an object to be in equilibrium, the sum of the clockwise moments must equal the sum of the counterclockwise moments. In this case, the man's shoulder is acting as the pivot point, and the weight of the ladder (200N) is creating a clockwise moment. To counteract this, the man must apply an equal and opposite counterclockwise moment with his arm pushing upwards (force F). By hanging a weight of 150N at a distance of 0.667m from the pivot point, the clockwise moment created by this weight will be equal to the clockwise moment created by the weight of the ladder, resulting in equilibrium and allowing force F to be 0N. Good job on using the equation for moment and solving for the distance!