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Fermi Questions: Weinstei@Physics.Odu.Edu Answers from November Larry Weinstein, Column Editor Old Dominion University, Norfolk, VA 23529; Fermi Questions: [email protected] Answers from November 10 Question 1, Neutron Stars: The Sun rotates on angular velocity must increase by 10 . This its axis once per month. If the Sun shrinks and becomes means that its period will decrease by 1010. Let’s a neutron star, by what factor will its rotational inertia express a month in SI units. 1 month = 30 days ϫ change? What will its new period of rotation be? (Note: 24 hr/day ϫ 60 min/hr ϫ 60 s/min ≈ 3 ϫ 106 s. the density of the Sun today is approximately Thus, the Sun’s new period will be T = 3 ϫ 106 s/ 103 kg/m3.) 1010 = 3 ϫ 10-4 s. Answer: First we need to know the density of a neutron star. To do that we need to estimate the Note that this method significantly overesti- density of neutronium. We can estimate the den- mates the change in rotational inertia because sity of neutronium in a few ways. If you know we assumed a uniform density. This is a good the size of the typical nucleus, you can extrapolate assumption for a neutron star but a very bad from that (remembering that 99.9% of the mass assumption for the Sun. Taking this into account of the atom is in the nucleus). The nucleus of lead properly should increase the neutron star period has a radius of 6 10-15 m so that its density is significantly. Copyright 2007, Lawrence Weinstein ρ = 0. 208 kg/mole 4 − π()610××15mat/mole 3 610 23 Question 2, Plastic Bags: People are concerned 3 that plastic bags waste resources. What is the total mass of all the grocery store plastic bags used by an average 17 3 ≈×410kg/m , American in one year? How does this compare with the mass of hydrocarbons burned in the average American’s automobile? or about 1015 times greater than the Sun. Answer: There are different ways to estimate the Alternatively, you can use the fact that a proton or mass of the average grocery store plastic bag. One neutron has a radius of about 10-15 m and work bag seems to have about the mass of one paper from that. Or, if you remember the atomic model clip or about 1 g; however, it is very difficult to where the nucleus is a marble on the 50-yard line compare the masses of such disparate items. The of a football field and the electron orbits cover stack of a hundred or so bags in the grocery store the entire field, then you would get an increase of has a mass of a few ounces (more than one ounce, density of (50 m/0.5 cm)3 = 1012, which is too less than a pound). At about 30 grams per ounce, small by a factor of 100 (but which does get the this also gives a mass per bag of about 1 g. We exponent correct to within 15%). can also estimate the volume. One bag is about 30 cm ϫ 30 cm. Estimating the thickness is more Since mass is conserved, radius r ρ-1/3. Thus, difficult. Those 100 bags in the grocery are about if the density increases by 1015, then the radius 1 cm thick (definitely more than 1 mm and less will decrease by a factor of 105. Rotational inertia than 10 cm) so each bag is about 10-4 m thick. is proportional to r 2 so that the Sun’s rotational This gives a volume per bag of V = 0.3 m ϫ inertia will decrease by a factor of 1010. Wow! 0.3 m ϫ 10-4 m = 10-5 m3. Since plastic has That is a lot. about the same density as water, this gives a mass of m = 10-5 m3 ϫ 103 kg/m3 = 10-2 kg = 10 g. In order to conserve angular momentum, if the Let’s take the geometric mean of our estimates and rotational inertia decreases by 1010, then the use m = 3 g. 452 THE PHYSICS TEACHER ◆ Vol. 45, 2007 Now we need to estimate the number of bags used ranties [and no, I don’t care about the difference per year. The average American family of four will between 10,000 and 15,000 here]) in a car or shop twice per week and buy 10 bags of groceries SUV that gets about 20 mi/gal (if your car gets each time. This means that they will use 2 ϫ 52 10 or 30 mi/gal, use the appropriate numbers). 10 = 1000 bags per year, or 250 bags per per- This means that she burns about V = 104 mi/yr/ son. This seems like a lot. However, the total mass 20 mi/gal = 500 gal/yr = 2 ϫ 103 L/yr. Since gas- of all those bags is only M = 3 g/bag 250 bags oline has about the density of water (for purposes = 0.8 kg ≈ 1 kg. Before we know whether this is a of this column), this means that she burns 2 ϫ little or a lot, we need to compare it to something 103 kg/yr of gasoline. else. Since plastic bags are made from hydrocar- bons, let’s compare this to the mass of hydrocar- Thus, the average American uses 2000 times less bons we burn in our cars. hydrocarbons in their grocery store plastic bags than in their automobile. Carefully keeping your The average American drives about 104 mi/yr tires properly inflated will save far more hydrocar- (since that is the average mileage used in car war- bons than avoiding plastic bags. Copyright 2007, Lawrence Weinstein THE PHYSICS TEACHER ◆ Vol. 45, 2007 453.
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