How does the math for lever forces work in this scenario?

[jackrabbit]

Point A
Point X 11 pounds
5 inches 5 inches ↑ 5 inches 5 inches
_________________________________________________________________
2 pounds 8 pounds
↓ ↓


Assume the horizontal line is a rod that is 20 inches long. The 11 pounds upward force is from a cable tied to the middle of the rod at Point A. There is a 2 pound downward force at left end (10 inches to the left of center), and an 8 pound downward force at 5 inches to the right of center. Point X is five inches to the left of center and is just a fixed point (imagine it is a rod sticking out, and that the horizontal rod runs into when it is lifted).

I am trying to figure out if the horizontal rod will rotate around point X, using the basic formula for force re a lever (i.e., the total force at the fulcrum is equal to force on the lever multiplied by distance to the fulcrum).

I experimented with this and found out that the horizontal rod dipped down on the end with 8 pounds. I was trying to understand the math.

My hypothesis, which was completely wrong, was as follows:

First, I thought of Point A as a fulcrum. Which would mean that the net force at this point was the difference between 8 x 5 and 2 x 10, i.e., a net downward force of 20

Then, turning to Point X, I thought the net force would be the difference between 11 x 5 (upward force) and 20 x 5 (downward force), which would mean a net upward force of 15. This would mean the rod would tilt up on the right side.

So, that was obviously wrong, and probably laughably so to those who actually understand this.

Can you please help me? How does the math here work?

 

[PhanthomJay]

Point A
Point X 11 pounds
5 inches 5 inches ↑ 5 inches 5 inches
_________________________________________________________________
2 pounds 8 pounds
↓ ↓


Assume the horizontal line is a rod that is 20 inches long. The 11 pounds upward force is from a cable tied to the middle of the rod at Point A. There is a 2 pound downward force at left end (10 inches to the left of center), and an 8 pound downward force at 5 inches to the right of center. Point X is five inches to the left of center and is just a fixed point (imagine it is a rod sticking out, and that the horizontal rod runs into when it is lifted).

I am trying to figure out if the horizontal rod will rotate around point X, using the basic formula for force re a lever (i.e., the total force at the fulcrum is equal to force on the lever multiplied by distance to the fulcrum).

This is an incorrect statement. The total moment (or torque) about the pivot is equal to the sum of the individual moments of the forces about the pivot. A clockwise moment may be considered as a negative moment, and a counterclockwise moment is then considered to be positive.

I experimented with this and found out that the horizontal rod dipped down on the end with 8 pounds. I was trying to understand the math.

My hypothesis, which was completely wrong, was as follows:

First, I thought of Point A as a fulcrum. Which would mean that the net force at this point was the difference between 8 x 5 and 2 x 10, i.e., a net downward force of 20

point A is not the fulcrum

Then, turning to Point X, I thought the net force

that should be net moment

would be the difference between 11 x 5 (upward force) and 20 x 5 (downward force), which would mean a net upward force of 15. This would mean the rod would tilt up on the right side.

So, that was obviously wrong, and probably laughably so to those who actually understand this.

Can you please help me? How does the math here work?

About the fulcrum point X, The 2 pound force produces a moment of 2(5) = 10, counterclockwise (ccw); the 11 pound force produces a moment of (11)(5) = 55 ccw; the 8 pound force produces a moment of (8)(10) = 80 clockwise (cw). So the net moment about the fulcrum is __________?? and it is cw or ccw? And I'm not laughing.

 

[jackrabbit]

Thanks. This is great.

So, 15 cw and the rod points down on the right side.

I had considered that possibility at one point but dismissed it since I thought point X could not be the fulcrum with respect to the 2 pound force since it is actually lifting the rod down off of point X, rather acting on point X directly. The only "fulcrum" (which I see now is not a fulcrum) it is acting directly on is point A. Well, that's what I get for making things overly complicated.

Thanks again for the clear and quick answer. Much appreciated.