Formula to determine effort to raise a ladder

[Davy Jones]

I am trying to find a formula to determine effort to raise a ladder based on the angle of the ladder at a given time. I am concerned mostly about the angle so I can plug in an individuals height and distance toward the fulcrum. I am assuming the ladder will not slide at the footing. I included a diagram that I hope will help. Ultimately, we are trying to decided between a 2 section 35' extension ladder and a 3 section will have a shorter length when bedded. Thank you.
http://www.arvideostorage.com/laddereffortquestion.jpg

 

[sk1105]

The amount of force you need to raise the ladder depends on how fast you want it to move, but for simplicity's sake we can assume you exert enough force to balance gravity. An infinitesimally small increase in force above this will raise the ladder, so you can consider this to be the minimum force required. Also important is how far away from the fulcrum the force is. These two quantities are combined in the calculation of moments.

Starting with gravity, recall that it always acts directly downwards, but to calculate moments, you will need to use trigonometry to calculate the component of gravity perpendicular to the ladder. Assuming the ladder is uniform along its length, you can assign a centre of mass (think about where that might be). This is the point from which all of the ladder's weight will act, at all times. To get the moment due to gravity, you just multiply the component of force perpendicular to the ladder by the distance along the ladder from the fulcrum to the centre of mass.

Now, to balance this with pushing. You can work out the distance along the ladder at which the person pushes using trigonometry with their feet, their hands and the fulcrum forming a right-angled triangle. This distance may not be the same as the distance at which gravity acts. It is not correct at this point just to make the forces equal; the moments must be equal. So if F is the pushing force and x is the distance of this force from the fulcrum, you will need to divide the moment you calculated earlier by x, and you have your force.

You'll find that x and the component of the gravitation force perpendicular to the ladder both depend on the angle, thereby introducing it into your final equation.

I've tried to give you mostly conceptual help to allow you to come up with the equations yourself, but if any step is unclear just leave a reply :)

 

[Davy Jones]

Based on the image can you provide some samples of how the formula should look.
1) If the moment of force is half way up the ladder at 10' and the individual is applying force at a height of 6' his feet would be 8' from the fulcrum with an angle at 37 degrees.
2) What would it then look like if the individual advanced 2' toward the fulcrum placing his feet 6' away. Now the moment of force would occur 8.5' up the ladder still at a height of 6' with a new angle of 44.5 deg? Appreciate the help.

 

[Svein]

In real life the ladder needs to have some down force at the point of contact with the ground, otherwise you will not be able to raise it.

• If you grab the ladder in the middle it will balance nicely but you will not be able to raise it.
  
• If you grab it at one end, you will be able to raise it, but not beyond your own reach.

So the answer must be somewhere between those two extremes...

 

[Davy Jones]

The amount of force you need to raise the ladder depends on how fast you want it to move, but for simplicity's sake we can assume you exert enough force to balance gravity. An infinitesimally small increase in force above this will raise the ladder, so you can consider this to be the minimum force required. Also important is how far away from the fulcrum the force is. These two quantities are combined in the calculation of moments.

Starting with gravity, recall that it always acts directly downwards, but to calculate moments, you will need to use trigonometry to calculate the component of gravity perpendicular to the ladder. Assuming the ladder is uniform along its length, you can assign a centre of mass (think about where that might be). This is the point from which all of the ladder's weight will act, at all times. To get the moment due to gravity, you just multiply the component of force perpendicular to the ladder by the distance along the ladder from the fulcrum to the centre of mass.

Now, to balance this with pushing. You can work out the distance along the ladder at which the person pushes using trigonometry with their feet, their hands and the fulcrum forming a right-angled triangle. This distance may not be the same as the distance at which gravity acts. It is not correct at this point just to make the forces equal; the moments must be equal. So if F is the pushing force and x is the distance of this force from the fulcrum, you will need to divide the moment you calculated earlier by x, and you have your force.

You'll find that x and the component of the gravitation force perpendicular to the ladder both depend on the angle, thereby introducing it into your final equation.

I've tried to give you mostly conceptual help to allow you to come up with the equations yourself, but if any step is unclear just leave a reply :)

Based on the image can you provide some samples of how the formula should look.
1) If the moment of force is half way up the ladder at 10' and the individual is applying force at a height of 6' his feet would be 8' from the fulcrum with an angle at 37 degrees.
2) What would it then look like if the individual advanced 2' toward the fulcrum placing his feet 6' away. Now the moment of force would occur 8.5' up the ladder still at a height of 6' with a new angle of 44.5 deg? Appreciate the help

 

[Davy Jones]

In real life the ladder needs to have some down force at the point of contact with the ground, otherwise you will not be able to raise it.

• If you grab the ladder in the middle it will balance nicely but you will not be able to raise it.
  
• If you grab it at one end, you will be able to raise it, but not beyond your own reach.

So the answer must be somewhere between those two extremes...

I get that but the ladder becomes increasingly more difficult to raise as the individual gets closer to the fulcrum until the center of gravity is in equilibrium

 

[berkeman]

How do you keep it vertical once it's raised? Is it tilted against something?

Can you have a 2nd person to the left, pulling on a rope attached to the top rung of the ladder? That will help with the efficiency and peak effort of raising the ladder.

 

[pbuk]

35 ft?? 150 lbs??! Show me the catalogue, I don't believe it.

 

[cpsinkule]

To make this easy let's just assume the ladder has uniform density ρ and it is fixed at one end to not slip or move and has length L. The potential energy for a point mass a distance h above the ground is U=mgh. therefore a little element of length along the ladder has potential energy ρgh*dr where dr is an element of length along the ladder. Now h for a given element is just rsinθ where r is the distance from the fixed point to the element and θ is the angle the ladder makes with the ground so ρgh*dr=ρgrsinθdr the potential energy of the entire ladder is just the integral of this from 0 to L ∫ρgrsinθdr=ρgsinθ*r²/2|₀^L=
ρgsinθL²/2 , but ρL=m so mgLsinθ/2=U the work you need to do to raise the ladder from an initial angle to a final angle is easily obtained here. It should be clear to you, from this formula, that the product mL is the deciding factor in which ladder takes the least amount of work to lift it a certain angle. Of course, this is only relevant if the ladder is fixed at one point and the mass density is uniform throughout each ladder.

 

[berkeman]

35 ft?? 150 lbs??! Show me the catalogue, I don't believe it.

Yeah, that does sound high. Here's a heavy 22' ladder that weighs just under 50 pounds...

http://www.lowes.com/pd_78463-287-MT-22_0__?k_clickID=613d57c8-4e92-b808-7b43-00006fc1db09&store_code=2842&productId=1101083&selectedLocalStoreBeanArray=%5Bcom.lowes.commerce.storelocator.beans.LocatorStoreBean%405fb85fb8%5D&storeNumber=2842&kpid=1101083&kpid=1101083&cm_mmc=SCE_PLA-_-LumberAndBuildingMaterials-_-LaddersFoldingStairsStepstools-_-1101083%3AWerner&DM_PersistentCookieCreated=true&CAWELAID=&CAWELAID=1210284624

 

[anorlunda]

Catalog reference here http://sunnyvale.ca.gov/Portals/0/Sunnyvale/DPS/Ladder%20-%20Specifications.pdf

Ladders for fire service duty are very strong and very heavy. It may need to be used as a bridge for the firefighter plus victim to cross, or as the high point anchor for a hoist.

It takes two men to raise such a ladder. One to hold the bottom down, and the other to walk it up. Maximum force for the walker happens at some mid distance. When the ladder is vertical, the walker needs zero lifting force. When I was the walker, it took all my strength.

 

[anorlunda]

Hmmm, the same site has a http://www.lowes.com/pd_65500-50116-D1540-2__?productId=1095741#BVRRWidgetID that weighs in at 90 lbs - that's a monster. I've never seen anything longer than 10 m (32'9") in the UK, but then we have the Working at Height Regulations.

Indeed yes. Next imagine the many volunteer fire companies who can't afford aluminum and still use wooden ladders.

 

[pbuk]

I removed that comment as it was not relevant to a fire and rescue situation.

 

[anorlunda]

I think the OP question was interesting. At the start, the walker lifts the far end of the ladder with a 2:1 mechanical advantage. Tha mechanical advantage goes to 1:1 at the mid point and less than one past the mid point. Meanwhile, as the ladder rises, more of it's weight is supported by the base, not the walker.

So what is the expression for the net force applied by the walker as a function of ladder angle? Ignore acceleration.

 

[Joel B]

Hey bud! I assume you want to know what the maximal effort needed will be. The maximal effort occurs when the ladder is at 45 degrees. I won't go into all the details, there's a great book on google that explains it. But because of that, the formula for max effort is (weight x bedded length) / 4H. H is the height of the person raising the ladder. I'll give an example of a ladder on our truck. It's a 45' 3 section extension ladder with Bangor poles. It weighs 242lbs and the bedded length is 19' (shorter than a 2 section 35!). (WxL)/(4xH) = (242x19)/(4x6) = 191.6lbs. That's a lot of weight to lift overhead! It's a pain, but we can do a two man raise. So much easier with 3 people though. As for a 35, you should have no issues with a 2 section 35. The Duo-Safety 35 is insanely light (compared to Alco-lite). It's 122lbs! So if you plug that into the formula for a 6 ft firefighter (and 20' bedded length) you get a max effort of 101.7lbs. I don't know any firefighter that can't push press 100lbs overhead all day, let alone for the few seconds it takes to get a ladder vertical. Even our Alco-lite 35's, which weigh 141lbs, only take a max effort of 117.5lbs. A very noticeable difference in effort in practice, but easily doable. Hope this helps!

For more on the math see here :
https://books.google.com/books?id=F...e the weight of a ladder being raised&f=false