1. How do we find the center of mass of a system?

2. Finding the Center of Mass by Integration
If the system has a continuous mass distribution, then we must using integration to find the center of mass. rcm = (1/M)∫rdm

3. Symmetry
The center of mass of a homogeneous, symmetric body must lie on the axis of symmetry.

4. 1. Uniform Rod
Where do you think the center of mass of a uniform rod is? We can write an equation for the linear mass density, λ = dm/dx. So dm = λdx AND xcm = 1/M ∫xdm

6. 2. Finding the Center of Mass of a Non-Uniform Rod
A rod of length 30.0 cm has a linear density (mass per length) given by λ=50.0g/m + (20.0g/m2)x
Where x is the distance from one end, measured in meters.
What is the mass of the rod?
How far from the x = 0 end is its center of mass?
M=15.9 g
x = cm

8. 3. Finding the Center of Mass of a Semicircular Hoop
Write down equation for center of mass. Write down an expression for dm. Write down an expression for x. Write down an expression for y.
dm=λdl = λrdθ

9. Center of Mass of Semicircular Hoop
rcm= 1/M∫(rcosϴi +rsinϴj)dm Substitute for dm and we get: rcm = R2/M ∫(cosϴi +sinϴj)λdθ The mass density… λ=M/πR Show that rcm=2R/πj