A vertical container (right rectangular prism) with base area measuring 0.15 m by 0.2 m is being filled with identical small stones, each with a volume of 2*10-8 m3 and a mass of 0.05*10-3 kg. Assume that the volume of the empty spaces between the stones is negligible. If the height of the stones in the container increases at the rate of 3*10-3 m/s, at what rate does the mass of the stones in the container increase?
The density of the stone is
ρ = m / V = 0.05*10-3 / 2*10-8 = 2500 kg/m3.
The total mass of the stones in the container when filled to height h is
M = ρ*V or M = ρ*A*h,
A = 0.15*0.2 = 0.03 m2
is the base area of the container that remains constant.
So, the rate of mass change is given by
dM/dt = d(ρ*A*h)/dt or
dM/dt = ρ*A*dh/dt (Since ρ and A are constant).
ρ*A*dh/dt = 2500 * 0.03 * 3*10-3 = 0.225 kg/s.