Determine the location of the center of mass of spheres

[emily081715]

Homework Statement

All three disks are made of sheet metal of the same material, and the diameters are 1.0 m , 2.0 m , and3.0 m . Assume that the x-axis has its origin at the left-most point of the left-most object and it points to the right. Part A) Determine the location of the center of mass of the system shown
.

part B)Repeat the calculation for three solid spheres all made of the same metal and having the same diameters as in part A.

Homework Equations

m1x1+m2x2+m3x3/m1+m2+m3

The Attempt at a Solution

i have gotten the answer for the first part with some help, the answer was 3.5m. i have tried repeating the calculations i took in part A. and got 3.5m again. this is what i did;
Masses
4πr²
mass 1: 4π(0.5)²=3.14159
mass 2:4π(1)²=12.56637
mass 3: 4π(1.5)²=28.2743

center of mass of the system
3.14159(0.5)+12.56637(1)+28.2743(4.5)/3.14159+12.56637+28.2743

 

[gneill]

Sphere mass goes as the volume, not the cross sectional area. How does the volume of a sphere vary with radius?

 

[emily081715]

Sphere mass goes as the volume, not the cross sectional area. How does the volume of a sphere vary with radius?

thank you i caught my error and got the correct answer of 3.8m

 

[gneill]

thank you i caught my error and got the correct answer of 3.8m

You're welcome.

Rather than pushing around all those digits you could do it symbolically. Let the first sphere's mass be M and have radius r (where r = 1/2 meter). The the masses of the spheres would be M, 8M, 27M (going as the radius cubed). Their center of mass locations would be r, 4r, and 9r. Plugging them into the center of mass formula:

$COM = \frac{(M)(r) + (8M)(4r) + (27M)(9r)}{M + 8M + 27M}$

$~~~~~~~~~~= \frac{276(r)(M)}{36M}$

$~~~~~~~~~~= \frac{23}{3}r$

And since r is 0.5 m, the result is $COM = \frac{23}{6}~m \approx 3.83~m$.