assignment Homework

Hockey

(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than \(1/r^2\) forces.

Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly circular bowl-shaped ice rink with radius \(a\). The central region of the bowl (\(r < 0.8a\)) is perfectly flat and the sides of the ice bowl smoothly rise to a height \(h\) at \(r = a\).

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance \(\frac{a}{2}\) from the center of the rink, is hit in such a way that it's initial velocity \(\vec v_0\) is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of \(\vec v_0\) does the puck just escape the rink?

assignment Homework

Effective Potentials: Graphical Version

Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

assignment Homework

Ice calorimetry lab questions

1. Plot your data I Plot the temperature versus total energy added to the system (which you can call \(Q\)). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.
2. Plot your data II Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The \(p\) subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.
3. Specific heat From your plot of \(C_p(T)\), work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?
4. Latent heat of fusion What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
5. Entropy of fusion The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where \(Q\) is the energy thermally added to the system and \(T\) is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?
6. Entropy for a temperature change Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where \(P(t)\) is the heater power as a function of time and \(T(t)\) is the temperature, also as a function of time.

assignment Homework

Free energy of a two state system

1. Find an expression for the free energy as a function of \(T\) of a system with two states, one at energy 0 and one at energy \(\varepsilon\).

2. From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.

3. Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.

4. Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).