Coffees and Bagels and Net Worth

    • assignment Heat shields

      assignment Homework

      Heat shields
      Stefan-Boltzmann blackbody radiation Thermal and Statistical Physics 2020 A black (nonreflective) sheet of metal at high temperature \(T_h\) is parallel to a cold black sheet of metal at temperature \(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.

      1. Solve for the net power transferred between the two sheets.

      2. A third black metal sheet is inserted between the other two and is allowed to come to a steady state temperature \(T_m\). Find the temperature of the middle sheet, and solve for the new net power transferred between the hot and cold sheets. This is the principle of the heat shield, and is part of how the James Web telescope shield works.

      3. Optional: Find the power through an \(N\)-layer sandwich.

    • assignment Power Plant on a River

      assignment Homework

      Power Plant on a River
      efficiency heat engine carnot Energy and Entropy 2021 (2 years)

      At a power plant that produces 1 GW (\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a temperature of \(500^{o}C\), and the waste energy is expelled into the environment at \(20^{o}C\).

      1. What is the maximum possible efficiency of this plant?

      2. Suppose you arrange the power plant to expel its waste energy into a chilly mountain river at \(15^oC\). Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 10 cents per kilowatt-hour?

      3. At what rate will the plant expel waste energy into this river?

      4. Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will the temperature of the river increase?

      5. To avoid this “thermal pollution” of the river the plant could instead be cooled by evaporation of river water. This is more expensive, but it is environmentally preferable. At what rate must the water evaporate? What fraction of the river must be evaporated?

    • assignment Scattering

      assignment Homework

      Scattering
      Central Forces 2023 (3 years)

      Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

      1. What is the initial angular momentum of the system?
      2. What is the initial total energy of the system?
      3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
      4. Sketch the effective potential.
      5. What is the angular momentum at \(r_{\rm{min}}\)?
      6. What is the total energy of the system at \(r_{\rm{min}}\)?
      7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
      8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
      9. What is the value of the effective potential at \(r_{\rm{min}}\)?
      10. For what values of the initial total energy are there bound orbits?
      11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

  • Energy and Entropy 2021 (2 years)

    In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

    Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

    In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.*

    Thus your savings \(S\) can be considered to be a function of your bagels \(B\) and coffee \(C\). In this problem we will also discuss the prices \(P_B\) and \(P_C\), which you may not assume are independent of \(B\) and \(C\). It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.

    1. The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of \(P_C\) and \(P_B\)?

    2. Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).

    3. Solve for the total differential of your net worth. Your net worth \(W\) is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.