Students are designing an experiment to demonstrate the conversion of mechanical
ID: 2004474 • Letter: S
Question
Students are designing an experiment to demonstrate the conversion of mechanical energy into thermal energy. They have designed the apparatus shown in the figure above. Small lead beads of total mass M and specific heat c (heat capacity is used in Q = mc?T) fill the lower hollow sphere. The valves between the spheres and the hollow tube can be opened or closed to control the flow of the lead beads. Initially both valves are open.a. The lower valve is closed and a student turns the apparatus 180° about a horizontal axis, so that the filled sphere is now on top. This elevates the center of mass of the lead beads by a vertical distance h. What minimum amount of work must the student do to accomplish this?
b. The valve is now opened and the lead beads tumble down the hollow tube into the other hollow sphere. If all of the gravitational potential energy is converted into thermal energy in the lead beads, what is the temperature increase of the lead?
c. The values of M, h, and c for the students' apparatus are M = 3.0 kg, h = 2.00 m, and c = 128 J/(kg · K). The students measure the initial temperature of the lead beads and then conduct 100 repetitions of the "elevate and drain" process. Again, assume that all of the gravitational potential energy is converted into thermal energy in the lead beads. Calculate the theoretical cumulative temperature increase after the 100 repetitions.
d. Suppose that the experiment were conducted using smaller reservoirs, so that M was one tenth as large (but h was unchanged). Would your answers to parts (b) and (c) be changed? If so, in what way, and why? If not, why not?
e. When the experiment is actually done, the temperature increase is less than calculated in part (c). Identify a physical effect that might account for this discrepancy and explain why it lowers the temperature.
Explanation / Answer
a. Thanks to the work-energy theorem we know that W = E, or that the change in energy of a system is equal to the work done to the system. Before we flip the apparatus, we know that there is no velocity and no height. This leads to us determining that there is originally no kinetic or potential energy, so the initial total energy is 0. After we flip the apparatus, there is now a potential energy due to the height of the lead beads. Potential energy is given by Mgh, with g being the gravitational acceleration at the surface of Earth. Kinetic energy is still zero, so the total energy is now Mgh. Since this change in energy is caused by the work done on the system, we know that work done by the student is also Mgh.
b. We know that the potential energy of the lead beads is Mgh. We also know that the the specific heat equation is Q = mcT. Since all of the potential energy is going to be converted into heat energy, we can set Mgh = McT. The M's cancel out and you simply solve for T.
c. Simply plug the given values into the equation we derived in part b and we will be able to solve for T. Then multiply this by 100 since this process will be repeated 100 times! Remember that g = 9.8 m/s2.
d. Realize that both potential energy (Mgh) and heat energy (McT) include a factor of M. What happens to each when M is changed by the same amount? (hint: what happens in the equation from part b when you change M?)
e. One effect that could effect the accuracy of this experiment is the property of conduction. As the lead beads lay in contact with the container, some of the heat energy will be transfered to the apparatus. This results in not all of the energy being used to heat the actual lead beads, so the actual temperature change will be less than calculated.