An initially empty beaker, in the shape of a cylinder by cross-sectional area A, is left out in the rain. The raindrops hit the beaker vertically downward with speed v. The rain continues at a constant rate so the height of the water in the beaker h(t) increases by time t at a ratedh/dt = w, where w is negligible compared to v. The raindrops rapidly come to rest inside the beaker, so we can neglect any kinetic energy of the water that has collected in the beaker. Let ρ denote the density of water (i.e. the mass per unit volume).
1. What is the rate at which the mass of the water in the beaker increases with time?
2. Let ycm(t) denote the height of the center of mass of all the water that has collected in the beaker by time t. What is the rated y cm/dt at which this height increases?
3. The total momentum of any system of particles Ptot is equal to the total mass Mtot times the velocity vcm of the center of mass. Should we conclude, therefore, that the water in the beaker has a vertical momentum equal to its mass times the value of dycm/dt as explained in (2)? Explain your answer in one or two sentences.
4. If the beaker is placed on a scale, while the beaker is still in the rain, the impact of the raindrops on the beaker will cause the reading of the scale to be larger than the weight of the beaker and the water it contains. By how much is the reading on the scale increased with the impact of the raindrops? (Neglect the effect of raindrops that hit the scale directly.)