Let a thin rod of length a have density distribution function ρ(x) = 10e^( -0.1x) , where x is measured in cm and ρ in grams per centimeter.
(a) If the mass of the rod is 30g, what is the value of a?
(b) For the 30g rod, will the center of mass lie at its midpoint, to the left of the midpoint, or to the right of the midpoint? Why?
(c) For the 30g rod, find the center of mass.
(d) At what value of x should the 30g rod be cut in order to form two pieces of equal mass? (Hint: Let z, with 0 ≤ z ≤ a, denote the point on the rod where it should be cut to form two pieces of equal mass, and set up an equality that will allow you to find z.)