Homework 2 Answers

1. Chapter 4, problem 5:
   1. A bowling ball has more mass than a baseball. Since gravitational potential energy depends on the mass of the object, for the same height above the ground, the bowling ball has more potential energy.
   2. Gravitational potential also depends on the height above the ground. A diver with more height has more potential energy.
   3. Finally, the other thing that the gravitational potential depends on is the mass of the planet (or star...). Jupiter, being more massive, makes objects in orbit have more gravitational potential energy.
2. 1. Since the seperation distance comes into the gravitational law is 1/d², if we triple the distance we have:
      
      Force_new/Force_old = (G M₁M₂/(3 d)²)/(G M₁M₂/(d)²)
      We can cancel a lot of stuff (G, M₁ and M₂) and re-arrange a bit:
      =d²/(3 d)²
      Distributing the squared:
      = d²/(d² 3²)
      The ds also cancel, leaving us with 1/9.
      
      So the gravitational force with 3 times the seperation is 9 times smaller.
   2. We worked this out in class (yay!). Jupiter is 318 times more massive than Earth, but 5 times farther from the Sun. So:
      Force_Jupiter/Force_Earth = (G M_SunM_Jupiter/(d_Jupiter)²)/(G M_SunM_Earth/(d_Earth)²)
      Cancelling what we can: = (M_Jupiter/(d_Jupiter)²)/(M_Earth/(d_Earth)²)
      Re-arranging
      = (M_Jupiter/M_Earth) (d_Earth/d_Jupiter)²
      I know that Jupiter is 318 times more massive than Earth and 5.2 times farther from the Sun, so:
      = (318) * (1/5.2)²
      =11.8
      
      So Jupiter exerts about 11.8 times more force on the Sun than the Earth does.
   3. If the Sun were magically (aliens?) replaced by a star twice as massive, one of the masses in Newton's Law of Gravity would be twice is large, making the force twice as large. (If you want to be careful about it, go ahead and do the ratio problem as I did above. It never hurts to be careful.)
3. Chapter 5, problem 13
   1. In this case, the orbit is exactly the same period. Remember, Kepler's Third Law only wants to know the average distance from the Sun! Mass of the planet doesn't matter!
   2. Newton's version of Kepler's Third Law says that p²=(2 π)² a³/(G M_star)
      
      p_new/p_old = ((2 π) a_new^3/2/(G M_new)^1/2)/((2 π) a_old^3/2/(G M_old)^1/2)
      Cancelling, again (a_{new=aold!) and re-arranging the Ms that are left.
      = (Mold/Mnew)^1/2
      = (1/4)^1/2
      = 1/2}
      So the planet's orbit is half of a year long.
4. Chapter 8, problem 18
   1. Let's see. Kepler's Third tells us that p²=a³. We can solve this for a: a=p^2/3. We need a in AU and p in years. We're given p in days, p= 4.23 days. It's easy to convert to years: 1 year=365.25 days, so 4.23 days * 1 year/365.25 days = 0.012 years. So we get a = (0.012 years)^2/3 = 0.05 AU. That's around ten times closer than Mercury!
   2. Since this is probably a jovian planet, it's unlikely that it formed so close to its star. Jovian planets need ices to form to their massive sizes, but hydrogen compouds were probably in their gasoues state that close to the star.
   3. Anything reasonably intelligent here is OK. The best guess right now is that the protoplanetary disk pulls back on the planet, making to move in towards the star. So you can form a planet where Jupiter is, then move it.