Directions: Questions 1, 4 and 5 are short free-response questions that require about 13 minutes each to answer and are worth 7 points each. Questions 2 and 3 are long free-response questions that require about 25 minutes each to answer and are worth 12 points each. Show your work for each part in the space provided after that part.
1.(7 points, suggested time 13 minutes)
Two blocks are connected by a string of negligible mass that passes over massless pulleys that turn with negligible friction, as shown in the figure above. The mass m2 of block 2 is greater than the mass m1 of block 1.The blocks are released from rest.
(a) The dots below represent the two blocks. Draw free-body diagrams showing and labeling the forces(not components) exerted on each block. Draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces.
(b) Derive the magnitude of the acceleration of block 2. Express your answer in terms of m1，m2 and g。
Block 3 of mass m3 is added to the system, as shown below. There is no friction between block 3 and the table.
(c) Indicate whether the magnitude of the acceleration of block 2 is now larger, smaller, or the same as in the original two-block system. Explain how you arrived at your answer.
2. (12 points, suggested time 25 minutes)
Some students want to know what gets used up in an incandescent lightbulb when it is in series with a resistor:current, energy, or both. They come up with the following two questions.
(1) In one second, do fewer electrons leave the bulb than enter the bulb?
(2) Does the electric potential energy of electrons change while inside the bulb?
The students have an adjustable power source, insulated wire, lightbulbs, resistors, switches, voltmeters,ammeters, and other standard lab equipment. Assume that the power supply and voltmeters are marked in 0.1 V increments and the ammeters are marked in 0.01 A increments.
(a) Describe an experimental procedure that could be used to answer questions (1) and (2) above. In your description, state the measurements you would make and how you would use the equipment to make them.Include a neat, labeled diagram of your setup.
i. Explain how data from the experiment you described can be used to answer question (1) above.
ii. Explain how data from the experiment you described can be used to answer question (2) above.
A lightbulb is nonohmic if its resistance changes as a function of current. Your setup from part (a) is to be used or modified to determine whether the lightbulb is nonohmic.
i. How, if at all, does the setup need to be modified?
ii. What additional data, if any, would need to be collected?
(d) How would you analyze the data to determine whether the bulb is nonohmic? Include a discussion of how the uncertainties in the voltmeters and ammeters would affect your argument for concluding whether the resistor is nonohmic.
3. (12 points, suggested time 25 minutes)
A block is initially at position x=0 and in contact with an uncompressed spring of negligible mass. The block is pushed back along a frictionless surface from position x=0 to x=-D,as shown above,compressing the spring by an amount △x=D. The block is then released. At x=0 the block enters a rough part of the track and eventually comes to rest at position x=3D。The coefficient of kinetic friction between the block and the rough track is μ.
(a) On the axes below, sketch and label graphs of the following two quantities as a function of the position of the block between x=-D and x=3D. You do not need to calculate values for the vertical axis, but the same vertical scale should be used for both quantities.
i. The kinetic energy K of the block
ii. The potentiael nergy U of the block-spring system
The spring is now compressed twice as much, to△x=2D. A student is asked to predict whether the final position of the block will be twice as far at x=6D. The student reasons that since the spring will be compressed twice as much as before, the block will have more energy when it leaves the spring, so it will slide farther along the track before stopping at position x=6D。
(b)i. Which aspects of the student’s reasoning, if any, are correct? Explain how you arrived at your answer.
ii. Which aspects of the student’s reasoning, if any, are incorrect? Explain how you arrived at your answer.
(c) Use quantitative reasoning, including equations as needed, to develop an expression for the new final position of the block. Express your answer in terms of D.
(d) Explain how any correct aspects of the student’s reasoning identified in part (b) are expressed by your mathematical relationships in part (c). Explain how your relationships in part (c) correct any incorrect aspects of the student’s reasoning identified in part (b). Refer to the relationships you wrote in part (c), not just the final answer you obtained by manipulating those relationships.
4. (7 points, suggested time 13 minutes)Two identical spheres are released
from a device at time t equals zero t=0 from the same height H, as shown
above.Sphere A has no initial velocity and falls straight down. Sphere B is
given an initial horizontal velocity of magnitude Vo and and travels a
horizontal distance D before it reaches the ground. The spheres reach the ground
at the same time tf,even though sphere B has more distance to cover before landing. Air resistance is negligible.
(a) The dots below represent spheres A and B. Draw a free-body diagram showing and labeling the forces (not components) exerted on each sphere at time tf/2
(b) On the axes below, sketch and label a graph of the horizontal component of the velocity of sphere A and of sphere B as a function of time.
(c) In a clear, coherent, paragraph-length response, explain why the spheres reach the ground at the same time even though they travel different distances. Include references to your answers to parts (a) and (b).
5. (7 points, suggested time 13 minutes)
The figure above shows a string with one end attached to an oscillator and the other end attached to a block. The string passes over a massless pulley that turns with negligible friction. Four such strings, A, B, C, and D, are set up side by side, as shown in the diagram below. Each oscillator is adjusted to vibrate the string at its fundamental frequency f. The distance between each oscillator and pulley L is the same, and the mass M of each block is the same. However, the fundamental frequency of each string is different.
(b) A student graphs frequency as a function of the inverse of the linear mass density. Will the graph be linear? Explain how you arrived at your answer.
(c) The frequency of the oscillator connected to string D is changed so that the string vibrates in its second harmonic. On the side view of string D below, mark and label the points on the string that have the greatest average vertical speed.