In a lab in my physics class, we were supposed to find the CG of my physics teacher. My partner and I had him lie flat on a wooden (assumed uniform) board that was resting on top of two PVC pipes at either end. Since the PVC pipes acted as the base of support, and since things balance when the CG is over the base of support, we pushed the pipes together, figuring that the point where the pipes met and the board balanced would be the CG of our teacher and the board combined.
The board was 2.445m long and had a mass of 11.7936kg.
His head was .378m from the designated top of the board and his feet were .142m from the bottom of the board. Our teacher is 1.925m tall and has a mass of 81.2kg.
The pipes met 1.204m from the bottom of the board.
We decided to attempt to solve this using torque, with counterclockwise as the positive direction. If you put the pivot point at where the two pipes met, then the torque equation looks like this (I think):
ΣT= (r1)(m(teacher)g)(sin90) – (r2)(m(board)g)(sin90) =0
r1 is the distance of the teacher's CG from the pivot point, and r2 is the distance of the board's CG from the pivot point. (r2 is the middle of the board, since we assumed the mass is uniformly distributed, so r2 = 2.445/2 = 1.2225m)
Using that information, I got that r1 = 0.17792069m. If you feel like checking my math, feel free.
So here's my question. With all of that information, how do I translate r1 into relevant terms and find the CG in relation to my teacher's height? In other words, how do I finish this sentence using the data I have?
"[Teacher]'s center of gravity is ______% of his height."
I really appreciate the help ahead of time!
To find the CG in relation to your teacher's height, you can use the concept of lever arms. The lever arm is the perpendicular distance between the pivot point (where the pipes meet) and the line of action of the force on an object. In this case, the forces are the weight of the board and the weight of your teacher.
Let's calculate the lever arms for each component:
1. Lever arm for the board:
Since the pipes meet 1.204m from the bottom of the board, the lever arm (r2) for the board is 1.204m.
2. Lever arm for the teacher:
The CG of the teacher is located at a distance of r1 = 0.17792069m from the pivot point (where the pipes meet). However, we need to calculate the lever arm for the teacher in relation to his height.
To do this, we can consider the triangle formed by the teacher's height (1.925m), the distance from the bottom of the board to his feet (0.142m), and the lever arm r1:
|\
| \
| \
h | \
| \
| \
| \
--------------|-------\
r1 | 0.142m
|
|
|
In this right triangle, we can use the Pythagorean theorem to find the length of the lever arm (r1'):
r1' = sqrt(r1^2 + h^2) = sqrt((0.17792069m)^2 + (1.925m)^2)
Now we have the lever arm r1' in terms of your teacher's height. We can use this information to find the CG in relation to your teacher's height.
To determine the CG percentage in relation to his height, we can use the formula:
CG percentage = (r1' / h) * 100
Substituting the values we calculated:
CG percentage = (sqrt((0.17792069m)^2 + (1.925m)^2) / 1.925m) * 100
Calculating this expression will give you the answer to complete the sentence:
"[Teacher]'s center of gravity is ______% of his height."