AP Physics 1 2015 Free Response Question 5

Question: 
A string is attached to an oscillator and its other end is attached to a block as shown below. The string passes over a pulley that is massless and frictionless. The distance between the oscillator and pulley is L and the mass of the block is M. The driving frequency of the oscillator is adjusted such that the string vibrates in its second harmonic. Identify locations on the string that have the greatest vertical speed.

Scoring Guidelines:

An indication that the string vibrates in its second harmonic or a wave is drawn such that λ2 = L.	1 point
Indicate points that are at the antinodes of any standing wave drawn on the string. (Full credit: the two points are located at one-fourth length of the string and three-fourths length of the string from the oscillator.)	1 point

(Source: http://apcentral.collegeboard.com/home)

Feynman's insights or goofs?:
The purpose of this question is to examine the properties of standing waves and apply the appropriate relationships among wavelength, frequency, and wave speed. Furthermore, students are expected to locate the nodes and antinodes of a string that is oscillating.

In this experiment, a string with one end is attached to an oscillator and its other end is attached to a block. Importantly, a node is not exactly located at the end that is attached to the oscillator because the oscillator vibrates up and down continuously. A definition of nodes is “[t]he points where there is no motion (Feynman et al., 1963, section 49-1 The reflection of waves).” If we observe the oscillations more carefully, the (virtual) node could be located slightly to the left of the oscillator. In a similar sense, an antinode does not occur exactly at the opening of the open pipe, but slightly beyond the opening where there is a lesser constraint for the air molecules to oscillate (See figure 1 below). The extra distance needed is sometimes called an end correction for the open-ended pipe. Thus, we should include the end correction for a more accurate calculation of the resonance frequency of the pipe. 

Fig 1

The scoring guidelines state that the two antinodes are located at one-fourth length of the string and three-fourths length of the string from the oscillator. Strictly speaking, the node does not occur exactly at the oscillator, and the antinodes can be located slightly to left of the one-fourth length of the string and three-fourths length of the string from the oscillator (See figure 2 below). Essentially, the points on the string that have the greatest average vertical speed can be located slightly to the left of the expected antinodes because of the end effect. Therefore, students could be penalized if the anti-nodes are not drawn exactly at the expected locations. Furthermore, some students might not indicate a node occurs at the oscillator which is not incorrect. 

Fig 2

In Feynman’s own words, “[s]uppose that the string is held at one end, for example by fastening it to an ‘infinitely solid’ wall. This can be expressed mathematically by saying that the displacement y of the string at the position x = 0 must be zero because the end does not move (Feynman et al., 1963, section 49–1 The reflection of waves).” However, there is a continuous vertical movement for the segment of the vibrating string attached to the oscillator. Thus, the node does not strictly occur at the location of the oscillator because the displacement of this segment of the string is not always zero. Furthermore, Feynman adds that “[t]he points where there is no motion satisfy the condition sin (ωx/c) = 0, which means that (ωx/c) = 0, π, 2π, …, nπ, … These points are called nodes (Feynman et al., 1963, section 49–1 The reflection of waves).” In short, the nodes should be defined as points that are always stationary.

Interestingly, Feynman mentions that “if we assume that the string is infinite and that whenever we have a wave going one way we have another one going the other way with the stated symmetry, the displacement at x = 0 will always be zero and it would make no difference if we clamped the string there (Feynman et al., 1963, section 49–1 The reflection of waves).” In a sense, Feynman has a slip of tongue when he simply says that “the string is infinite.” To be precise, the vibrating string is not infinitely long or massive, but that it is attached to an “infinitely rigid” wall. Thus, physics teachers could explain that Feynman is sometimes sloppy in his language usage.

More importantly, the fundamental frequency of the vibrating string is not simply dependent on the linear mass density and tension in the real world. Feynman explains that “[t]he idea that the natural frequencies are harmonically related is not generally true. It is not true for a system of more than one dimension, nor is it true for one-dimensional systems which are more complicated than a string with uniform density and tension. A simple example of the latter is a hanging chain in which the tension is higher at the top than at the bottom (Feynman et al., 1963, section 49–3 Modes in two dimensions).” Nevertheless, the stiffness of a real-life string can cause the wave velocity to be also dependent on the wavelength (Elmore & Heald, 1985).

Note: 

1. A node can be distinguished as a displacement node or a pressure node. For example, a “pressure node (corresponding to a displacement or velocity antinode) occurs at the open end of a tube, while a pressure antinode (corresponding to a displacement or velocity node) occurs at the closed end (Gregersen, 2011, p. 37).”

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=z_KX8Xpxa-c

References:

1. Elmore, W. C., & Heald, M.A. (1985). Physics of Waves. New York: Dover.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
3. Gregersen, E. (2011). The Britannica Guide to Sound and Light. New York: The Rosen Publishing Group.

AP Physics 1 2015 Free Response Question 4

Question: 
Two identical spheres emerge from an electronic device at the same time (t = 0) and from the same height (h = H), as shown below. Sphere A has no initial velocity and moves downward in a straight line. Sphere B moves with an initial horizontal velocity (v₀) and covers a horizontal distance D when it hits the ground. The two spheres reach the ground at the same time (t_f) though sphere B moves parabolically and travels a longer distance before landing. (Assume there is no air resistance.)

Students have to explain why the two identical spheres reach the ground at the same time if they are dropped simultaneously.

Scoring Guidelines: 

Indicate the horizontal motions of the two spheres do not affect their vertical motions.	1 point
Indicate the two spheres move with the same vertical velocity.	1 point
Indicate the two spheres move with the same vertical acceleration.	1 point
Indicate the two spheres falling from the same height would take the same time.	1 point
There is/are no incorrect or irrelevant statement(s).	1 point

(Source: http://apcentral.collegeboard.com/home)

Comments:
The purpose of this question is to assess students’ understanding of two-dimensional motion, as well as equations behind one sphere that is dropped from rest and a second identical sphere that is projected with an initial horizontal speed from the same height. In this post, we discuss a part of the question which is about why a sphere that is falling vertically in a straight line can reach the ground at the same time as the other sphere that is falling in a parabolic path. Importantly, the explanation of this phenomenon could include theoretical and empirical knowledge.

1. Theoretical idealization: Based on the scoring guidelines, a point can be awarded if students indicate that the two spheres falling the same height, would take the same time. That is, the horizontal velocity of the sphere does not affect its vertical velocity. In other words, the horizontal motion and vertical motion are independent of each other. However, we have assumed that the falling of an object is independent of its mass. Strictly speaking, mass independence is not valid for an observer on the ground (Lehavi & Galili, 2009). By using Newton’s law of gravitation or Newton’s third law, the Earth can accelerate toward the falling object. Thus, the statement that “all objects fall to the ground with the same acceleration” is a theoretical idealization and it is only approximately correct.

2. Empirical knowledge: Based on the scoring guidelines, another point can be awarded if students indicate that the difference in horizontal motion does not affect the vertical motion of the spheres. However, we should not simply deduce the independence of horizontal and vertical motions to be a theory. Similarly, we should not assume that two objects of different mass will reach the ground at the same time by simply using a thought experiment. Importantly, the falling of two spheres to the ground can be verified by experiments. Note that the air resistance acting on the two spheres are in different directions while they are falling toward the ground. Furthermore, it is empirically verified that the frictional drag on an object is proportional to the velocity or proportional to the square of the velocity depending on how fast the object is moving.

Nevertheless, if Einstein were a student taking this examination, he might pose the following thought experiment: “I stand at the window of a railway carriage which is traveling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the ‘positions’ traversed by the stone lie ‘in reality’ on a straight line or on a parabola? (Einstein, 1961, p. 10).” In essence, the so-called straight line or parabolic path is dependent on the velocity of the observer. Simply phrased, we can explain the object’s motion in terms of the observer’s frame of reference. Perhaps, based on the scoring guidelines, Einstein or some students could be penalized in answering this question by using the concept of inertial reference frame? 

Feynman’s insights and goofs?:
We can understand the question from the perspectives of empirical knowledge and theoretical idealization.
1. Empirical knowledge: Feynman explains that “[a] falling body moves horizontally without any change in horizontal motion, while it moves vertically the same way as it would move if the horizontal motion were zero. In other words, motions in the x-, y-, and z-directions are independent if the forces are not connected. (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).” Essentially, the motions of an object can be resolved into perpendicular directions that are independent of each other. Furthermore, in the Fig. 7-3 of Volume I of The Feynman Lectures on Physics, there is an apparatus that demonstrate the independence of vertical and horizontal motions (Feynman et al., 1963). Thus, the independence of the motions of the object in the x-, y-, and z-directions can also be considered as an empirical fact.

2. Theoretical idealization: Feynman writes that “[w]hat happens if we shoot a bullet faster and faster? Do not forget that the earth’s surface is curved. If we shoot it fast enough, then when it falls 16 feet it may be at just the same height above the ground as it was before. How can that be? It still falls, but the earth curves away, so it falls ‘around’ the earth (Feynman et al., 1963, section 7–4 Newton’s law of gravitation).” In other words, we have idealized the ground to be flat if the object falls within a short distance. This approximation does not hold if the object moves at a relatively high horizontal speed. In short, there are theoretical idealization and approximations involved in the falling of the object. The earth is not perfectly flat.

Note:

1. If Galileo were a student taking this examination, he might explain that the two spheres reach the ground at the same time because we have assumed that the air resistance is negligible. For example, in the Fourth Day of Dialogues Concerning Two New Sciences, Galileo (1638) writes that “if we consider only the resistance which the air offers to the motions studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety in the form, weight, and velocity of the projectiles (p. 252).”

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=x7UHl3Umeeg

References: 

1. Einstein, A. (1961/1916). Relativity: The Special and The General Theory. New York: Random House.

2. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Galileo, G. (1638/1954). Dialogues Concerning Two New Sciences (trans. H. Crew & A. de Salvio). New York: Dover.

4. Lehavi, Y., & Galili, I. (2009). The status of Galileo’s law of free-fall and its implications for physics education. American Journal of Physics, 77(5), 417-423.

AP Physics 1 2015 Free Response Question 3

Question: 
A block is initially stationary at position x = 0 and is in contact with a massless spring that is uncompressed. The block is then slowly compressed along a frictionless surface from position x = 0 to x = -D (See figure 1 below) such that ∆x = D. When the block is released, it reaches a rough part of the track at the right-hand side of x = 0 and it eventually slows down to rest at position x = 3D. Assume the coefficient of kinetic frictional force between the block and rough track is μ.

 

                                                                                 Fig. 1

In this question, students have to explain the correct and incorrect aspects of the following statement: “if the spring is compressed twice as long as before, the block has more energy when it leaves the spring, so it will slide farther along the track before stopping at position x = 6D.”

Scoring Guidelines:

Identify the correct aspect: the block has more energy when it leaves the spring.	1 point
Identify the incorrect aspect: the new final position of the block is not at x = 6D. (The spring’s elastic potential energy is proportional to the square of ∆x.)	1 point

(Source: http://apcentral.collegeboard.com/home)

Comments:

The main purpose of this question is to examine the relationship between the energy stored in a compressed spring-block system and the work done by the frictional (kinetic) force on the block after it leaves the spring. In other words, the question is about the transformation of the elastic potential energy of the block into its kinetic energy and thermal energy (microscopic internal energy). However, there are terminological and language issues in this question.

1. Terminological issues: It is surprising that the question and scoring guidelines adopt imprecise terminologies. For example, based on the scoring guideline for (b)(i), the student is correct because the block will have more energy when it leaves the spring. Note that the term “energy” could be changed to “kinetic energy.” To be more precise, it could be further changed to “translational kinetic energy” that is different from “rotational kinetic energy.” Next, based on the scoring guideline for (b)(ii), the student is incorrect because the spring’s energy does not scale linearly with its compression. However, the term “spring’s energy” could be changed to “elastic potential energy.” It is stated in AP 1: Algebra-Based Exam Description that, “Essential knowledge 4.C.1: The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy.”

2. Language issues: The sentence “the block will have more energy when it leaves the spring” in the question is misleading. Firstly, it seems to suggest that the block will have more energy, and thus violating the conservation of energy. More importantly, one may expect the block to be in contact with the spring until the spring is no longer compressed. Thus, after the block leaves the spring, the block’s (kinetic) energy starts to decrease at a constant rate because of frictional force of the track. This contradicts the statement that “the block will have more energy when it leaves the spring.” Perhaps the scoring guidelines could recognize students who identify the incorrect reasoning pertaining to “the block will have more energy when it leaves the spring”? Or perhaps the sentence could be rephrased as “the block will have relatively more kinetic energy when there is an increase in the compression of the spring”?

Feynman’s insights or goofs?:
The question adopts the term “spring’s energy” instead of “elastic potential energy.” In Feynman’s words, “[e]lastic energy is the formula for a spring when it is stretched. How much energy is it? If we let go, the elastic energy, as the spring passes through the equilibrium point, is converted to kinetic energy and it goes back and forth between compressing or stretching the spring and kinetic energy of motion. There is also some gravitational energy going in and out, but we can do this experiment ‘sideways’ if we like (Feynman et al., 1963, section 4–4 Other forms of energy).” That is, the motion of a spring may involve elastic energy, kinetic energy, and gravitational energy. However, physics teachers can use the following terms that are more precise: elastic potential energy, translational kinetic energy, and gravitational potential energy.

On the other hand, this question assumes that the frictional force is constant. Conversely, Feynman explains that “the frictional drag on a ball or a bubble or anything that is moving slowly through a viscous liquid like honey, is proportional to the velocity, but for motion so fast that the fluid swirls around (honey does not but water and air do) then the drag becomes more nearly proportional to the square of the velocity (Feynman et al., 1963, section 12-2 Friction).” Furthermore, Feynman states that “the drag force on an airplane is approximately a constant times the square of the velocity, or F ≈ cv² (Feynman et al., 1963, section 12-2 Friction).” However, the drag force can also be approximated by F ≈ av + bv². Importantly, Feynman clarifies that “as we study this law of the drag on an airplane more and more closely, we find out that it is ‘falser’ and ‘falser’ (Feynman et al., 1963, section 12-2 Friction).”

Interestingly, Feynman does not mention that the frictional force is independent of contact area which can be found in many physics textbooks. He explains that when there is good contact between two solids, they can hold very tight together and thus have a larger frictional force. Most important, Feynman adds that “to a fairly good approximation, the frictional force is proportional to this normal force, and has a more or less constant coefficient; that is, F = μN, where μ is called the coefficient of friction. Although this coefficient is not exactly constant, the formula is a good empirical rule for judging approximately the amount of force that will be needed in certain practical or engineering circumstances. If the normal force or the speed of motion gets too big, the law fails because of the excessive heat generated. It is important to realize that each of these empirical laws has its limitations, beyond which it does not really work (Feynman et al., 1963, section 12-2 Friction).”

Note:

1. Feynman also explains that “[i]n experiments of the type described above, the friction is nearly independent of the velocity. Many people believe that the friction to be overcome to get something started (static friction) exceeds the force required to keep it sliding (sliding friction), but with dry metals, it is very hard to show any difference (Feynman et al., 1963, section 12-2 Friction).” 

2. Jonathan Thomas-Palmer, for example, strongly disagrees with the use of the symbol K for the kinetic energy of the block because k stands for spring constant in the same question. (Source: https://www.youtube.com/watch?v=6dlijzUQHw4)

3. For another discussion of this question, you can visit the following websites:

https://www.youtube.com/watch?v=qQM-IGnxi6g
https://www.youtube.com/watch?v=KH60uz5LNWk

Reference: 

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

AP Physics 1 2015 Free Response Question 2

Question: 

Students are expected to describe an experimental procedure that could be used to answer the following question: “do fewer electrons leave a light bulb than enter the bulb in one second?” Next, they have to explain how experimental data can be used to answer the abovementioned question. (Assume that students are provided with voltmeters that are marked in 0.1 V increments and ammeters that are marked in 0.01 A increments.)

Scoring Guidelines:

Describe an analytical method of using experimental data and explain how the data can be used to answer the abovementioned question. Example: If the electric current measured is the same on both sides of the light bulb, then the number of electrons entering the bulb is the same as those leaving the bulb in one second.

1 point

(Source: http://apcentral.collegeboard.com/home)

Comments: 
In this question, students are expected to have a good concept of series circuits. The main purpose of the question is to assess students’ knowledge of experimental design and their ability in data analysis by using ammeters and voltmeters. Students were also asked to account the uncertainty of electrical measurement and to discuss how it affects the experimental results.

In the first part of this question, two ammeters are connected in series on both sides of the bulb to measure the electric current entering and leaving the bulb (based on the scoring guidelines). Importantly, we should not conclude that “the number of electrons per second entering and leaving the bulb is the same” if the electric current is measured to be the same on both sides of the bulb. There are both theoretical and experimental issues in this question. 

1. Theoretical issues: If the electric current through the ammeters on both sides of the bulb are the same, then the student expects the number of electrons per second entering and leaving the bulb are the same. This is based on a “simplified” definition of electric current: the number of electrons per second. However, to quote Feynman, “[e]lectric currents are electrons or other charges in motion with a net drift or flow (Feynman et al., 1964, section 13-2 Electric current; the conservation of charge).” Firstly, the electric current could be contributed by a flow of positive charge carriers such as copper atoms in the wire, but they are “almost essentially” stationary. Secondly, the magnitude of electric current is also dependent on the drift velocity of electrons. Thus, the electric current is not simply the number of electrons per second moving through the bulb.

2. Experimental issues: In this question, the ammeters are marked in 0.01 A increments. However, one may prefer to use clamp meters which have 0.001 A resolution and they do not introduce additional electrical resistance. More importantly, if the uncertainty in measuring an electric current is 0.01 A, this may imply a possible error of 0.01 C in one second. Therefore, the uncertainty in measurement corresponding to the number of electrons per second could be of the order 10¹⁷ because the electric charge of an electron is 6.02 × 10^-19 C. In a sense, this experiment is analogous to the use of a meter ruler to conclude that the atoms and electrons have the same length. From an empiricist’s perspective, one should not comment about the number of electrons per second entering and leaving the bulb are the same when there are limitations (or significant uncertainty) in measuring the electric current accurately.

Currently, an ampere is defined as the electric current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these two conductors a magnetic force that is equal to 2 × 10^–7 N/m. In other words, the ampere is not defined as one electron at a particular time based on current technology. Furthermore, the definition of an ampere in terms of two infinitely long conductors cannot be precisely implemented in practice. Recently, Dr. Giblin (2016) and researchers from National Physical Laboratory’s Quantum Detection Group and the University of Cambridge have reported the counting of electrons trapped in an electron pump (a small region of a specially designed semiconductor device) with a fidelity of one part per million. Thus, the electron pump could be the new primary reference standard of electric current. However, there are still significant errors in counting the number of electrons.

Feynman’s insights or goofs?:

Feynman states that “[e]lectric currents are electrons or other charges in motion with a net drift or flow (Feynman et al., 1964, section 13–2 Electric current; the conservation of charge).” In other words, an electric current is a net drift or flow of charge carriers such as electrons. However, Feynman’s definition of the electric current can be improved. For example, physics teachers may prefer to define the electric current as the time rate of flow of charge carriers. That is, we can distinguish the rate of flow with respect to time from the rate of flow with respect to displacement. Furthermore, the electric current can be more comprehensively defined as the rate of flow of ‘free’ electrons (or other charge carriers) per unit time due to a potential difference across the ends of an electrical conductor, under constant circuit conditions.

On the other hand, some physicists advocate the use of operational definitions. However, Feynman did not provide an operational definition of electric current. For example, Karplus (2003) writes that: “Operational definition: Electric current is measured by the dial reading of a standard ammeter (p. 315).” Furthermore, an ampere is defined as the electric current that can produce an attractive force of 2 × 10⁻⁷ N per meter of length  between two straight, parallel conductors of infinite length placed one meter apart in a vacuum. Importantly, a problem of the operational definition of electric current is that an uncertainty of 0.01 ampere in measuring an electric current corresponds to an uncertainty of 10¹⁷ electrons per second. Note that this operational definition is a measure of an effect of electric current that is a magnetic force.

Interestingly, Feynman has an insightful explanation on the effects of electric current: “[w]e ask what happens in a piece of resistance wire when it is carrying a current. Since the wire has resistance, there is an electric field along it, driving the current. Because there is a potential drop along the wire, there is also an electric field just outside the wire, parallel to the surface. There is, in addition, a magnetic field which goes around the wire because of the current. The E and B are at right angles; therefore there is a Poynting vector directed radially inward, as shown in the figure. There is a flow of energy into the wire all around. It is, of course, equal to the energy being lost in the wire in the form of heat. So our “crazy” theory says that the electrons are getting their energy to generate heat because of the energy flowing into the wire from the field outside. Intuition would seem to tell us that the electrons get their energy from being pushed along the wire, so the energy should be flowing down (or up) along the wire. But the theory says that the electrons are really being pushed by an electric field, which has come from some charges very far away, and that the electrons get their energy for generating heat from these fields (Feynman et al., 1964, section 27-5 Examples of energy flow).”

Note:
1. You may want to visit this website:

http://feynman-answer.blogspot.sg/2016/08/electric-current-flow-of-electrons-or.html

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=OpbYvr8yhqQ

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley. 

2. Giblin, S. P., See, P., Petrie, A., Janssen, T. J. B. M., Farrer, I., Griffiths, J. P., ... & Kataoka, M. (2016). High-resolution error detection in the capture process of a single-electron pump. Applied Physics Letters, 108(2), 023502.
3. Karplus, R. (2003). Introductory physics: A Model approach (2nd ed.). Buzzards Bay: Captain’s Engineering Services.

AP Physics 1 2015 Free Response Question 1

Question: 
This question is based on a simple Atwood’s machine: two blocks are connected by a massless string passing over massless pulleys which are frictionless (See figure 1 below). Block 2’s mass (m₂) is greater than block 1’s mass (m₁). Students have to derive the magnitude of the acceleration of block 2 when the two blocks are released from rest. The answer should be expressed in terms of m₁, m₂, and g.

Fig. 1

It could be of interest to mention that a similar question can be found in Feynman’s Tips on Physics. The question involves the calculation of frictional force: “Two masses, m₁ = 4 kg and m₃ = 2 kg, are connected with cords of negligible weight over essentially frictionless pulleys to a third mass, m₂ = 2 kg. The mass m₂ moves on a long table with a coefficient of friction…(Feynman et al., 2006, p. 144).”

Scoring Guidelines:

Using Newton’s second law for block 1 m1a = T – m1g	1 point
Using Newton’s second law for block 2 m2a = m2g – T	1 point
Eliminate T and derive the acceleration: T = m1a + m1g m2a = m2g – m1a – m1g (m2 + m1)a = (m2 – m1)g a = (m2 – m1)g / (m1 + m2)	1 point

(Source: http://apcentral.collegeboard.com/home)

Comments:

The main purpose of the question is to assess students’ ability to apply Newton’s second law of motion to a modified Atwood’s machine. Students are expected to have a basic knowledge of free body diagrams and an understanding of factors that determine the acceleration of the system. However, they may have difficulty with the use of conventions for force and acceleration because one block is moving upward and the other block is moving downward.

We may expect this question to be simply answered by using Newton’s Second law of motion as shown below:

To find the acceleration of block 2, we can visualize an “equivalent question” in which the forces acting on block 1 and block 2 are shown in figure 2 below: Fig. 2 The larger force, m2g, acting on block 2 is the driving force and the smaller force, m1g, acting on block 1 is the opposing force. For block 2: the driving force (m2g) is in the direction to the right, whereas the tension (T) is in the direction to the left. By using Newton’s Second Law of motion, ΣF = ma we have m2g – T = m2a ------ (1) (Convention: The forces that are in the direction to the right are assigned positive.) For Block 1: The tension (T) is in the direction to the right and the opposing force (m1g) is in the direction to the left. By using Newton’s Second Law again, we have T – m1g = m1a ------(2) By solving the two equations (1) and (2), a = (m2 – m1)g / (m1 + m2) However, we can visualize block 1 and block 2 as a system. In this case, the total mass is m1 + m2 and the external forces on the system are m2g – m1g. Thus, a = F/m = (m2 – m1)g / (m1 + m2) Would students be penalized by using only one equation?

Feynman's insights?:
Alternatively, Feynman explains that “the laws of Newton could be stated not in the form F = ma but in the form: the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).” That is, it is possible to use Euler-Lagrange’s equation to answer this question. Interestingly, Mr. Bader introduced the principle of least action to his student, Feynman. In addition, Dias, Araújo, Silva, Santos, Barros, & Carvalho-Santos (2012), for example, propose to introduce Euler-Lagrange’s equation in introductory physics. However, it is unclear whether students who use a more advanced method could be awarded extra credit or penalized instead. (I happen to know of a high school student who was able to apply Euler-Lagrange’s equation to solve mechanics problems that are even more difficult.)

It is not difficult to use Euler-Lagrange’s equation to solve this question on Atwood’s machine. It can be accomplished by using only a few steps as shown below: 
The kinetic energy of the system (T) is given by T = ½ m₁ẋ² + ½ m₂ẋ² 

The potential energy of the system (U) is given by U = – m₂gx – m₁g(l – x) 

Thus, the Lagrangian can be written as 

T – U = ½m₁ẋ² + ½m₂ẋ² + m₂gx + m₁g(l – x)

By using Euler-Lagrange’s equation of motion, d/dt(∂L/∂ẋ) – ∂L/∂x = 0

We get (m₁ + m₂)ẍ – (m₂ – m₁)g = 0

Therefore, ẍ = (m₂ – m₁)g/(m₁ + m₂)

(Would some physics teachers penalize students for using Euler-Lagrange’s equation to get the correct answer?)

Note:
1. In Feynman’s words, “when I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare this lecture I found myself making more analyses on the thing. Instead of worrying about the lecture, I got involved in a new problem. The subject is this—the principle of least action (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).”

2. For another discussion of this question, you can visit the following websites:
https://www.youtube.com/watch?v=NrNG6OPijeY
https://www.youtube.com/watch?v=ff6SNrgUFt0

References:
1. Dias, C. F., Araújo, M. A., Silva, G. M., Santos, C. A., Barros Jr, P., & Carvalho-Santos, V. L. (2012). Adaptation of the Euler-Lagrange equation for studying one-dimensional motions in a constant force. arXiv preprint arXiv:1209.2197.

2. Feynman, R. P., Gottlieb, M. A., Leighton, R. (2006). Feynman's tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.