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.
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.”
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.
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1 point
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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.)
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1 point
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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).”
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-IGnxi6ghttps://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.
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