How do you find the center of mass for an object with multiple parts?

[PhysicForFun]

Homework Statement

1. Find of the center of mass of the image below.

2. A human head weighs 8lbs, a human torso weighs 20lbs, and a human's hips and legs weighs 20lb. The headis 10m long, the torso is 25m long, and the hips and legs are 35m long. Find the center of mass of the body.

Homework Equations

X = Ʃ(m1x1...) / Ʃ(m1...)
Y = Ʃ(m1y1...) / Ʃ(m1...)

The Attempt at a Solution

Sorry I couldn't find a clear example. I have two different methods given to solve these problems, so I wouldlike clarification.

1. X = (2kg)(2m) + (.5kg)(5m) / (2kg + .5kg) = 2.6m
Y = (2kg)(2m) + (.5kg)(1m) / (2kg + .5kg) = 1.8m
So this one took the problems into point masses. Then solved for them.

Spoiler

X1= (2kg)(0m) + (2kg)(4m) / (2kg + 2kg) = 2m
Y1 = (2kg)(0m) + (2kg)(4m) / (2kg + 2kg) = 2m
X2 = (.5kg)(4m) + (.5kg)(6m) / (.5kg + .5kg) = 5m
Y2 = (.5kg)(0m) + (.5kg)(2m) / (.5kg + .5kg) = 1m

2. X = (8lb)(10m) + (20lb)(25m) + (20lb)(35m) / (8lb + 20lb + 20lb) = 26.6m

Why does it just use the lengths instead of combining them, or even taking the half-way point? Maybe I misinterpretted the problem? I assumed this at first.

X = (8lb)(5m) + (20lb)(17.5m) + (20lb)(52.5m) / (48lb) = 30m
But I guess those shapes can't be divided perfetcly in half. But why is it not
X = (8lb)(10m) + (20lb)(35m) + (20lb)(70m) / (48lb) = 45m
This clearly is too big to be logical.

Could someone explain this? Or link me to some CoM problems that aren't using point masses. I think thepoint masses are sort of self-explanatory, but I can't find good examples otherwise.

Thanks.

 

[ehild]

You get the centre of mass by multiplying the mass of a piece with the distance of the CM of that piece from a reference point. As nothing has been specified about the shape of the pieces you can asume that their CM is at the middle. If you measure the distance from the head-top of the human,

X = (8lb)(5m) + (20lb)(17.5m) + (20lb)(52.5m) / (48lb) = 30m

is correct, but it means the distance from the same reference point.

By the way, what unit is "m"?

 

[PhysicForFun]

meter, but clearly that was wrong. It was probably inches lol.

So the answer page that has the solution above should just be ignored? I mean it also seems like a plausible answer. The math just doesn't make sense.

 

[ehild]

See: http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html
Calculating CM by pairs is not wrong, but overcomplicates the problem.

ehild

 

[cupid.callin]

X = (8lb)(5m) + (20lb)(17.5m) + (20lb)(52.5m) / (48lb) = 30m

Why is it 17.5 and nor 22.5 ??

 

[ehild]

You are right, it was a mistake. So X=(8lb)(5m) + (20lb)(22.5m) + (20lb)(52.5m) / (48lb) = 32m

ehild

 

[PhysicForFun]

Ok thanks. That math makes a lot more sense then the given answer.