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

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

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

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

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 = .1528 cm

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

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