How much did it really weigh? (leverage question)

[Tog_]

Let's say that you have a heavy thing in roughly the shape of a cube. We'll give it a base of 1 meter by 1 meter and a height of about 1.25 m. Mass is 750 Kg.

Now, let's say a person were to lift one side of this cube about 3 cm using the other side of it as a fulcrum. How much weight was actually lifted and how would you go about finding it out using math that a person that had 2 tries at algebra I would understand?

[HenrikOlsen]

A problem with your question is that "how much weight was lifted" isn't clearly defined, since you could be asking "How much energy was used to lift it?" or "That is the weight experienced by the lifter once the cube is lifted 3 cm?" which will give slightly different answers, since the first asks about the average weight experienced during the lift while the other asks about the weight experienced once the thing is lifted, both of which will be slightly less than 375kg, with the latter being slightly less than the former.

A complete answer for question two can be thought of as "If I raise my end a very tiny bit, how much is the center of gravity raised?" If you take the ratio between those two distances and multiply by the total weight you get how much weight you experience.

Check the wonderful HenrikDoodle(tm) attached here.
Angle A is how much the block is turned when you've lifted the edge 3 cm, it's asin(3cm/1m) (http://www.google.com/search?q=asin%283cm%2F1m%29+in+degrees%3F)=1.719 degrees.
Angle B is the angle between the base and line from the fulcrum to the center of gravity, it's atan(1.5m/1m) (http://www.google.com/search?q=atan%281.5m%2F1m%29+in+degrees%3F)=56.310 degrees.
Angle A+B (http://www.google.com/search?q=asin%283cm%2F1m%29%2Batan%281.5%29+in+deg rees%3F)= 58.029 degrees is then the angle between horizontal and the line from the fulcrum to the center of gravity.

Now, turning the block a teeny tiny bit q will raise the edge by H=1m*cos(A)*q, while raising the center of gravity by h=sqrt((3/4m)2+(1/2m)2)*cos(A+B)*q, so calculate h/H=sqrt((3/4m)2+(1/2m)2)/1m*cos(A+B)/cos(A) (http://www.google.com/search?q=cos%28asin%283cm%2F1m%29%2Batan%281.5%29% 29%2Fcos%28asin%283cm%2F1m%29%29*sqrt%28%283%2F4%2 9%5E2%2B%281%2F2%29%5E2%29%3F)= 0.477489868 and multiply by 750kg and you get that you're lifting 358.117401kg.

[Neverfly]

By lift- do you two mean push the top edge?:neutral:
The word lift is throwing me for a loop here.

It's actually an easy equation that I can't remember at the momenthttp://us.i1.yimg.com/us.yimg.com/i/mesg/emoticons7/9.gif
Looking at my bookshelf now...

[pghnative]

To expand on Henrik's point, you could also (with some difficulty) figure out how much force you need to apply to hold the cube steady at any point in the rotation. Two equations need to be solved simultaneously, one easy, one reasonly easy with some trig.

Eq 1: F(tog) + F(fulcrum) = W
Eq 2: F(tog)* L(tog) = F(fulcrum) * L(f)

where
F(tog) is your force (obviously),
F(fulcrum) is the force from the fulcrum (ground)
W = weight of cube
L(t) is the horizontal distance from the center of mass to the point that Tog lifts
L(fulcrum) is the horizontal distance from the center of mass to the point of the fulcrum

The point of Eq 1 is obvious -- tog and the fulcrum exactly support the weight of the cube

The point of eq 2 is that there is no net force of rotation. The torque applied by tog must equal the torque applied by the fulcrum for the cube to not rotate. (Note: The way I've written the equation implies a positive number for the distance.)

So some trig is needed to determine the horizontal distance of the center of mass from the fulcrum and tog, but otherwise the math is pretty simple.

And obviously these equations assume no acceleration (ie, you hold the cube in position).

[Tog_]

By lift- do you two mean push the top edge?:neutral:
The word lift is throwing me for a loop here.

Lift. being under and grunting.:lol:
Henrick's little ascii diagram had it dead on.

It was a coke pallet stacked on top of another one. I stood with my back to it and put my fingers in the pallet where the forks go, then straightened my legs. I know how much the thing weighed, and I had a good idea how strong I was. It came up in a conversation this morning at work and I got to wondering just how much I actually lifted. I figured it was about half, but I wasn't sure if there was other stuff I wasn't taking into consideration.

And here's where I start to over think stuff. If each side is supporting 1/2 the weight, why would getting a longer arm for the lever make a difference? Shouldn't the distance from the fulcrum to the force matter?

Bah. Forum hung up on me so I didn't see the reply with the math in it. I t might answer the above question after I have a chance to actually understand it.

[pghnative]

By lift- do you two mean push the top edge?:neutral:
The word lift is throwing me for a loop here.

You can use the same equations for this situation as shown in my previous post, though in this case one needs to be more rigorous and split F(tog) into horizontal and vertical components. Need to do this for both equation 1 and equation 2. In equation 2, each needs to be multiplied by either the horizontal distance or the vertical distance from the center of mass. Signs (positive or negative) need to be dealt with carefully.

For the case of pushing the top edge, the equations show that the fulcrum and you each must push with a force equal to the weight of the cube. The reason is that Tog doesn't have any vertical force, so the fulcrum holds all the weight of the cube. By eq 2, Tog must push with an equal force (each are same distance from the COM) to hold the cube just off the ground. So pushing requires twice the force as lifting. (Though pushing is likely easier on the back than lifting!)

[pghnative]

If each side is supporting 1/2 the weight, why would getting a longer arm for the lever make a difference? You're not really using a lever, though.

[HenrikOlsen]

Lift. being under and grunting.:lol:
Henrick's little ascii diagram had it dead on.
Talk about overthinking, I thought the 3 cm mattered so I revised my answer:)

It made for a difference of 16.88 kg in the answer where "about half" would have been close enough.

[Neverfly]

Talk about overthinking, I thought the 3 cm mattered so I revised my answer:)
Right. I was looking up some equations simply because the higher you lift- you tip to a point where the weight pulls in your favor and much less force is needed.

[HenrikOlsen]

And here's where I start to over think stuff. If each side is supporting 1/2 the weight, why would getting a longer arm for the lever make a difference? Shouldn't the distance from the fulcrum to the force matter?
It's the ratio of the length of the lever to the distance between the center of gravity and the fulcrum that matters, double the baseline and you move the center of gravity twice as far away too, keeping the ratio constant.

[Tog_]

Right. I was looking up some equations simply because the higher you lift- you tip to a point where the weight pulls in your favor and much less force is needed.

Not with coke pallets. Oh they tip to that point, then the high side gets heavy again.
And sticky.
And loud.
Then the swearing starts.

[HenrikOlsen]

You're not really using a lever, though.
It is actually considered a lever even though he's lifting at the same side as the weight is pushing down, the math is identical to that of handling a classical lever, you just count both distance to fulcrum and weight as negative, and since they are multiplied in the formulae you can't see a difference between lifting on one side and pushing down in the other.

[01101001]

Is there a furniture mover in the house?

[Neverfly]

Not with coke pallets. Oh they tip to that point, then the high side gets heavy again.
And sticky.
And loud.
Then the swearing starts.

That's why I prefer Dr pepper.

When those tip over, you end up with a quiet setting in which everyone is busy on hands and knees- tongue to the floor.

[Tog_]

That's why I prefer Dr pepper.

When those tip over, you end up with a quiet setting in which everyone is busy on hands and knees- tongue to the floor.

I don--

Sorry. Threw up in my mouth a little there.

I don't know if you've ever really looked at the backroom floor in a grocery store. Especially in front of the soda section.

The worst tipping incident we had involved 5 total pallets in three stacks.
And one set of ceiling lights.
And what we genuinely believed were white meat department coats.

Here's a tip for you. If you ever come across a 12 or 24 pack with clear tap on one end. Don't get it. Some, very likely angry, merchandiser spent an additional 2 or 3 hours taping up the survivors after a tipping incident. Either that or it was stuff that was in seasonal packaging and it's a couple weeks after the season. In the first case, they are not always really careful about getting all the right flavors in the cases, and usually put them in sticky.