Monday April 14

We talked about linear density along a rod, and mass as the integral of density.
Using the linear density, you can compute not just the total mass (integral of rho(x) dx),
but also the rotational inertia (integral of x rho(x) dx).

We also talked about population density. If the density per square mile is only a function of the distance from the origin, then total population = integral_r=0^r=D rho(r) 2 pi r dr, where the extra factor of 2 pi r accounts for the length of the circle at radius r.

Then we talked about change in energy as
integral of (force(t) times speed(t) times cos(angle between force and velocity) dt).
So if the force is pushing the same direction as the moving object, it gains energy;
if it's the opposite direction, it loses energy;
if it's perpendicular (like whirling something around your head on a string) it stays the same.