Simple Nature - Light and Matter

proportion to one another, so this has no effect on the conclusion.)Conservation of energy does not, however, tell us anything obviousabout which ball gets there first. This is a general problemwith applying conservation laws: conservation laws don’t refer directlyto time, since they are statements that something stays thesame at all moments in time. We expect on intuitive grounds thatthe ball that goes by the lower ramp gets to B first, since it buildsup speed early on.Buoyancy example 10⊲ A cubical box with mass m and volume V = b 3 is submergedin a fluid of density ρ. How much energy is required to raise itthrough a height ∆y?⊲ As the box moves up, it invades a volume V ′ = b 2 ∆y previouslyoccupied by some of the fluid, and fluid flows into an equal volumethat it has vacated on the bottom. Lowering this amount of fluidby a height b reduces the fluid’s gravitational energy by ρV ′ gb =ρgb 3 ∆y, so the net change in energy is∆E = mg∆y − ρgb 3 ∆y= (m − ρV )g∆y .In other words, it’s as if the mass of the box had been reduced byan amount equal to the fluid that otherwise would have occupiedthat volume. This is known as Archimedes’ principle, and it istrue even if the box is not a cube, although we’ll defer the moregeneral proof until page 203 in chapter 3. If the box is less densethan the fluid, then it will float.A simple machine example 11⊲ If the father and son on the seesaw in figure k start from rest,what will happen?⊲ Note that although the father is twice as massive, he is at halfthe distance from the fulcrum. If the seesaw was going to startrotating, it would have to be losing gravitational energy in order togain some kinetic energy. However, there is no way for it to gainor lose gravitational energy by rotating in either direction. Thechange in gravitational energy would bej / How much energy is requiredto raise the submergedbox through a height ∆y?k / A seesaw.∆U = ∆U 1 + ∆U 2= g(m 1 ∆y 1 + m 2 ∆y 2 ) ,but ∆y 1 and ∆y 2 have opposite signs and are in the proportion oftwo to one, since the son moves along a circular arc that coversthe same angle as the father’s but has half the radius. Therefore∆U = 0, and there is no way for the seesaw to trade gravitationalenergy for kinetic.l / The biceps muscle is areversed lever.Section 2.1 Energy 85