Forces and Potential Energy - Middle School Physical Science

1. A book is moved in three different ways while Earth's gravity acts downward: 1) lifted straight up by $1\,\text{m}$, 2) carried horizontally across a room at the same height, 3) lowered straight down by $1\,\text{m}$. Which choice correctly describes how gravitational potential energy changes in each case (ignoring air resistance)?
   Answer: Up: increases; Horizontal: stays the same; Down: decreases. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): gravitational force points downward because gravitational PE is lower at ground level (h=0, PE=0) than at height (h>0, PE=mgh>0), so gravity points toward the minimum PE position; moving upward (against force) increases PE, downward (with force) decreases PE, and horizontally (perpendicular to force) keeps PE the same since height doesn't change. For gravitational PE changes: lifting up 1 m increases height, so PE increases (ΔPE = mg*1 m >0); horizontal carry keeps height same, so PE stays the same (Δh=0, ΔPE=0); lowering down 1 m decreases height, so PE decreases (ΔPE = -mg*1 m <0). Choice B is correct because it properly connects the movements to PE changes: up increases, horizontal stays the same, down decreases. Choice A is wrong because it confuses the directions: claims up decreases PE when up actually increases PE (against gravity), and horizontal increases when it stays the same (no height change). The force-PE relationship helps predict motion and understand energy: only vertical movements change gravitational PE, as it's height-dependent. Systematic approach: identify if motion changes height (up: +Δh, increases PE; horizontal: Δh=0, same PE; down: -Δh, decreases PE), and recognize no change horizontally because force is vertical.
2. A ball is on a smooth hill. Point P is at the top of the hill (highest point), and Point V is in the valley (lowest point). If the ball is released from rest near the top, it rolls toward the valley. Which statement best connects force direction and potential energy (PE)?
   Answer: The net force points toward lower PE, so the ball accelerates "downhill" toward V where gravitational PE is smaller. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): gravitational force points downward because gravitational PE is lower at ground level (h=0, PE=0) than at height (h>0, PE=mgh>0), so gravity points toward the minimum PE position. For gravitational force on a hill: at top P (highest), PE maximum, at valley V (lowest), PE minimum, and the net force (component of gravity) points downhill toward V (lower PE), so released ball accelerates toward V, decreasing PE and increasing KE. Choice B is correct because it correctly identifies net force points toward lower PE, so ball accelerates downhill to V. Choice A reverses force-PE by saying toward higher PE (uphill), Choice C wrongly claims force always upward (gravity down), and Choice D says force unrelated to PE (but directly connected). The force-PE relationship helps predict motion and understand energy: objects move toward PE minimum if free, converting PE to KE. Real examples: water flows downhill toward lower gravitational PE.
3. A horizontal spring is attached to a wall. The spring is at its relaxed length at Position R, and it can be compressed to Position C1 (compressed $5\,\text{cm}$) or Position C2 (compressed $10\,\text{cm}$). The spring force on the attached block points away from the wall (toward the relaxed length) when the spring is compressed. Which statement is correct about force direction and elastic potential energy (PE)?
   Answer: Elastic PE is lowest at the relaxed position, and the spring force points toward the relaxed position (toward lower PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): for springs, the force points toward the relaxed position because elastic PE is lower there (PE=0 at natural length) than when compressed or stretched (PE=½kx²>0). For spring force and elastic PE: A compressed spring has elastic potential energy stored (PE > 0 when compressed), and the spring force points outward (away from compressed position, toward the relaxed natural length position where PE = 0 minimum), so from C2 (10 cm compression, higher PE) or C1 (5 cm, lower but still >0), the force points toward R (relaxed, minimum PE). Choice C is correct because it properly connects that elastic PE is lowest at relaxed, and the force points toward this lower PE position. Choice A misidentifies the minimum PE as most compressed (actually maximum PE there), Choice B reverses by saying force points toward more compression (higher PE), and Choice D incorrectly claims PE doesn't change with compression. The force-PE relationship helps predict motion and understand energy: compressing against the force increases PE, while releasing allows motion in force direction, decreasing PE and increasing KE. Real examples: a compressed spring snaps back toward relaxed (lower PE), demonstrating forces point toward lower energy, with systems naturally moving toward minimum PE if free to do so (converting PE to KE).
4. A student lifts a backpack of mass $5\,\text{kg}$ from the floor to a shelf $1.5\,\text{m}$ high at constant speed. Take $g \approx 10\,\text{N/kg}$. Which statement correctly describes the work and the change in gravitational potential energy of the backpack?
   Answer: The student does work against gravity, and the backpack's gravitational potential energy increases by about $75\,\text{J}$. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): gravitational force points downward because gravitational PE is lower at ground level (h=0, PE=0) than at height (h>0, PE=mgh>0), so gravity points toward the minimum PE position; when you move an object in the force direction (falling downward in direction of gravity), the force does positive work and potential energy decreases (PE converts to KE typically), but when you move opposite to the force (lifting upward against gravity), you do work against the force and potential energy increases (your work is stored as PE = mgh), with the connection W = ΔPE for conservative forces like gravity. For gravitational force and PE in this lifting scenario: the student moves the backpack upward (opposite to downward gravitational force), doing work against gravity equal to mgh = 5 kg 10 N/kg 1.5 m = 75 J, which increases the backpack's gravitational PE by 75 J (from floor h=0, PE=0 to shelf h=1.5 m, PE=75 J). Choice C is correct because it properly connects that the student does work against gravity, increasing the backpack's gravitational potential energy by about 75 J. Choice D is wrong because it suggests work by gravity increases PE when actually gravity does negative work during lifting (opposing the motion), and the PE increase comes from the student's work against gravity. The force-PE relationship helps predict motion and understand energy: lifting against gravity stores energy as increased PE, which can later be released as KE if the object falls. Real examples: like raising a book to a shelf requires work against gravity, increasing PE, similar to how water pumped uphill stores gravitational PE for later use in hydroelectric power.
5. Two objects have the same positive electric charge. They are held a small distance apart and then released. Which statement best matches the force direction and the change in electric potential energy (PE) as they move?
   Answer: The electric force pushes them apart toward lower PE, so PE decreases as the distance increases. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases: for two positive charges that repel, PE is high when close together and low when far apart, so the repulsive force points outward (toward greater separation). For electric potential energy with two positive charges: Two positive charges repel each other (electric force pushes them apart), and the PE relationship shows: close together (small distance), high electric PE (repelling charges forced close have energy stored—like compressed spring), far apart (large distance), low electric PE (separated naturally is lower energy for repulsion), and force points from high to low PE (force pushes apart, from close/high-PE toward far/low-PE). When released, they move in the force direction (apart), so: the repulsive force does positive work pushing them apart, electric PE decreases as separation increases (PE converts to KE), and they accelerate away from each other as stored PE is released. Choice B is correct because it accurately states that the electric force pushes them apart toward lower PE and that PE decreases as distance increases. Choice A incorrectly states the force pulls them together when like charges repel (push apart); Choice C incorrectly claims electric force is always downward when electric forces act along the line between charges; Choice D incorrectly claims there's no force when like charges always repel with a force. The force-PE relationship explains repulsion: like charges naturally move apart (toward lower PE), converting stored electric PE into kinetic energy as they accelerate away from each other.
6. A student lifts a $3\,\text{kg}$ textbook from a desk ($h=1\,\text{m}$) to a shelf ($h=3\,\text{m}$). Take $g\approx 10\,\text{N/kg}$. If the lift is slow so the book's speed doesn't change much, what is the change in gravitational potential energy, and what does that say about the work done against gravity?
   Answer: $\Delta PE = 60\,\text{J}$, so $60\,\text{J}$ of work is done against gravity (stored as PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases: when you move an object opposite to the force (lifting upward against gravity), you do work against the force and potential energy increases (your work is stored as PE = mgh). For this specific problem: Initial height h₁ = 1 m, so initial PE = mgh₁ = (3)(10)(1) = 30 J; final height h₂ = 3 m, so final PE = mgh₂ = (3)(10)(3) = 90 J; therefore ΔPE = final PE - initial PE = 90 - 30 = 60 J. Since the book is lifted slowly (constant speed, no acceleration), the net force is zero, meaning the lifting force equals the gravitational force (3 × 10 = 30 N downward). The student does work against gravity: W = force × distance = 30 N × 2 m = 60 J upward, and this 60 J of work done against gravity is stored as the 60 J increase in gravitational PE. Choice A is correct because it accurately calculates ΔPE = 60 J and correctly explains that this represents work done against gravity that is stored as PE. Choice B incorrectly states ΔPE = -60 J when PE increases (positive change) during lifting; Choice C incorrectly calculates ΔPE = 6 J (likely forgetting to multiply by g); Choice D incorrectly claims ΔPE = 0 when there's clearly a height change that changes PE. The force-PE relationship helps us understand energy storage: lifting requires energy input (work against gravity), and this energy is stored as gravitational PE, ready to be released if the book falls back down.
7. Two small objects have the same positive electric charge. They are held at two different separations: Position A: $2\,\text{cm}$ apart, Position B: $8\,\text{cm}$ apart. For two like charges (repelling), which statement correctly describes the electric force direction and the electric potential energy change as they move from Position A to Position B?
   Answer: The electric force pushes them apart, and electric potential energy decreases as they get farther apart. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph); for electric forces, two positive charges repel each other (electric force pushes them apart), and the PE relationship shows: close together (small r), high electric PE (repelling charges forced close have energy stored—like compressed spring), far apart (large r), low electric PE (separated naturally is lower energy for repulsion), so force points from high to low PE (pushes apart, from close/high-PE toward far/low-PE). For electric potential energy with like charges: at Position A (2 cm apart), PE is higher (k q1 q2 / 0.02, large since small r), at Position B (8 cm apart), PE is lower (k q1 q2 / 0.08, smaller), and the repulsive force pushes them apart (toward increasing separation, decreasing PE). Choice B is correct because it correctly identifies that the electric force pushes them apart, and electric potential energy decreases as they get farther apart from A to B. Choice C is wrong because it states PE increases as they get farther apart, when for repelling like charges, PE decreases with separation (lower energy when apart). The force-PE relationship helps predict motion and understand energy: released close together, repelling charges accelerate apart, converting PE to KE. Real examples: like-charged balloons push away from each other, moving toward lower PE as they separate, demonstrating forces pointing toward lower energy.
8. A horizontal spring is fixed to a wall. Its elastic potential energy is $0\,\text{J}$ when the spring is relaxed (not compressed or stretched). A student compresses the spring and holds it at three different compressions: Position A: $0\,\text{cm}$ (relaxed), Position B: $2\,\text{cm}$ compressed, Position C: $4\,\text{cm}$ compressed. Which statement correctly describes the spring force direction and the elastic potential energy at these positions?
   Answer: The spring force points toward the relaxed position, and elastic potential energy is highest at Position C. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): for springs, elastic force points toward the relaxed position because elastic PE is minimum there (PE=0 at x=0) and higher when compressed or stretched (PE=½kx²>0). For spring force and elastic PE: a compressed spring has elastic potential energy stored (PE > 0 when compressed), and the spring force points outward (away from compressed position, toward the relaxed natural length position where PE = 0 minimum); at Position A (0 cm, relaxed), PE=0 (minimum), at B (2 cm compressed), PE=½k(0.02)² >0, at C (4 cm compressed), PE=½k(0.04)² (highest, four times B since squared). Choice B is correct because it correctly identifies that the spring force points toward the relaxed position, and elastic potential energy is highest at Position C. Choice A is wrong because it reverses the force-PE relationship: claims force points toward more compression (higher PE) when actually points toward relaxed (lower PE), and misidentifies lowest PE at C when it's highest there. The force-PE relationship helps predict motion and understand energy: a compressed spring, if released, expands toward the relaxed position, converting PE to KE. Systematic approach: identify force direction (outward toward relaxed), determine PE at positions (increases with compression), recognize force points toward decreasing PE, and predict release from C moves toward A, decreasing PE.
9. Two identical positive charges are held a small distance apart. Because they repel, the electric force on each charge points away from the other charge (toward larger separation). If the charges are released from rest and move farther apart, what happens to the electric potential energy of the two-charge system?
   Answer: It decreases because the electric force does work as they move apart, and moving in the force direction lowers potential energy. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): gravitational force points downward because gravitational PE is lower at ground level (h=0, PE=0) than at height (h>0, PE=mgh>0), so gravity points toward the minimum PE position. When you move an object in the force direction (falling downward in direction of gravity), the force does positive work and potential energy decreases (PE converts to KE typically), but when you move opposite to the force (lifting upward against gravity), you do work against the force and potential energy increases (your work is stored as PE = mgh). The connection between work and PE change is: W = ΔPE (work equals the change in potential energy for conservative forces like gravity, elastic, and electric forces). For electric potential energy: Two positive charges repel each other (electric force pushes them apart), and the PE relationship shows: close together (small distance), high electric PE (repelling charges forced close have energy stored—like compressed spring), far apart (large distance), low electric PE (separated naturally is lower energy for repulsion), and force points from high to low PE (force pushes apart, from close/high-PE toward far/low-PE). If you move them from small to smaller distance (closer together, against the repulsive force that pushes apart), you do work against the electric force, increasing electric PE (closer = higher PE for repelling charges, work stored as PE). If you release them at small distance, electric force does work pushing them apart (from small toward large separation), decreasing electric PE and increasing their KE (they accelerate apart as PE → KE). Choice B is correct because it properly connects that electric potential energy decreases as the electric force does work moving them apart in the force direction, lowering PE. Choice A reverses PE change: moving in force direction decreases (not increases) PE. The force-PE relationship predicts repelling charges separate, converting PE to KE. Opposite charges snap together (electric force toward lower PE for attraction).
10. A student compares the same $m$-kg object lifted straight up by $1\,\text{m}$ on Earth and on the Moon. The object's gravitational potential energy change is $\Delta U = mg\Delta h$. Earth: $g \approx 10\,\text{N/kg}$; Moon: $g \approx 1.6\,\text{N/kg}$. Which statement is correct?
    Answer: The change in potential energy is larger on Earth because $g$ is larger; the potential energy increases more per meter of height where the force is stronger. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): gravitational force points downward because gravitational PE is lower at ground level (h=0, PE=0) than at height (h>0, PE=mgh>0), so gravity points toward the minimum PE position. When you move an object in the force direction (falling downward in direction of gravity), the force does positive work and potential energy decreases (PE converts to KE typically), but when you move opposite to the force (lifting upward against gravity), you do work against the force and potential energy increases (your work is stored as PE = mgh). The connection between work and PE change is: W = ΔPE (work equals the change in potential energy for conservative forces like gravity, elastic, and electric forces). For gravitational PE: ΔU = m g Δh; stronger g on Earth means larger force, so larger ΔU (more work against stronger force to lift 1 m). Choice C is correct because it states ΔPE is larger on Earth due to larger g, as PE increases more per meter where force is stronger. Choice A reverses: weaker g on Moon means smaller ΔPE, less work to lift. Force magnitude relates to PE slope: steeper on Earth (stronger force). Objects weigh more on Earth, store more PE when lifted.
11. A block attached to a spring is pulled to the right, stretching the spring, and then released. The spring force on the block points left (back toward the relaxed position). As the block moves left (in the direction of the spring force), what happens to the spring's elastic potential energy (PE)?
    Answer: Elastic PE decreases because the spring force does work as the spring returns toward its relaxed (lower-PE) length. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): for springs, the force points toward the relaxed position because elastic PE is lower there (PE=0 at natural length) than when stretched (PE=½kx²>0). For spring force and elastic PE: when stretched to the right (PE > 0), the spring force points left toward relaxed (minimum PE), and as the block moves left (in force direction) after release, the spring does positive work, decreasing elastic PE (converting to KE as it returns to relaxed). Choice B is correct because it properly explains that elastic PE decreases as the block moves in the force direction toward relaxed (lower PE), with the force doing work. Choice A reverses by saying PE increases when moving in force direction (actually decreases), Choice C claims PE constant (but compression/stretch changes PE), and Choice D confuses by requiring outside agent for decrease (but release allows force to decrease PE). The force-PE relationship helps predict motion and understand energy: releasing a stretched spring causes motion in force direction, decreasing PE and increasing KE. Real examples: a rubber band snaps back when released, converting elastic PE to KE as it moves toward lower PE.
12. A student compresses a spring by pushing a block inward from the relaxed position to a more-compressed position. The spring pushes outward on the block. Which statement best describes the work and potential energy change during the compression (ignoring friction)?​
    Answer: The student does work against the spring force, and the spring's elastic potential energy increases by the amount of work put in. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that when you move an object against a force (compression moves inward against outward spring force), you do work against the force and potential energy increases (your work is stored as PE). For this spring compression scenario: the spring force points outward (trying to push block back to relaxed position), the student pushes inward (opposite to spring force), so the student does work against the spring force. This work against the force increases elastic PE: starting at relaxed position (PE = 0), ending at compressed position (PE = ½kx² > 0), so ΔPE = ½kx² - 0 = ½kx² (positive, PE increases). The work-energy relationship states: work done against force = increase in PE, so student's work input W = ΔPE = ½kx². Choice A is correct because it correctly identifies that the student does work against the spring force (pushing inward against outward force) and accurately states that elastic PE increases by the amount of work put in (W = ΔPE). Choice B incorrectly claims the spring does work on the student during compression—the spring does negative work (opposes motion) while the student does positive work; Choice C incorrectly states no work is done when work against force is definitely done during compression; Choice D incorrectly limits elastic PE to stretching only, when PE = ½kx² applies to both compression and extension. The systematic approach confirms: identify force direction (spring pushes out), identify motion direction (student pushes in), recognize motion against force requires work input, and this work increases PE by the same amount.
13. A ball sits in a smooth bowl-shaped track. The lowest point of the bowl is position X. A higher point on either side is position Y. Which statement best explains why position X is a stable equilibrium in terms of force direction and potential energy (PE)?
    Answer: Position X is stable because it has minimum PE; if the ball is moved to Y, the force points back toward X (toward lower PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that stable equilibrium occurs at PE minimum because forces point toward lower PE, causing displaced objects to return. For the bowl-shaped track: Position X at the bottom has minimum height, so gravitational PE = mgh is minimum there. Position Y on either side has greater height, so PE is higher at Y than at X. When the ball is at Y, the component of gravitational force along the track points down the slope toward X (toward lower PE). If the ball is displaced slightly from X to either side, the force points back toward X (the PE minimum), causing the ball to roll back—this defines stable equilibrium. The ball naturally seeks the lowest PE position: released from Y, it rolls toward X (converting PE to KE), passes through X with maximum speed (minimum PE, maximum KE), climbs the other side (converting KE back to PE), and oscillates around the stable equilibrium at X. Choice B is correct because it accurately identifies X as having minimum PE and explains that forces point back toward X (toward lower PE) when the ball is displaced. Choice A incorrectly claims maximum PE makes equilibrium stable when actually minimum PE creates stability; Choice C incorrectly states forces point toward higher PE; Choice D incorrectly claims PE doesn't depend on position in a gravitational field. The force-PE relationship defines stability: stable equilibrium occurs at PE minima where surrounding forces point inward, while unstable equilibrium occurs at PE maxima where surrounding forces point outward.
14. A student carries a backpack at a constant height across a flat hallway. The backpack moves horizontally while gravity pulls downward. What happens to the backpack's gravitational potential energy (PE) during this horizontal motion, and why?
    Answer: PE stays the same because the height doesn't change (motion is perpendicular to gravity). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that gravitational PE = mgh depends only on vertical height h, not on horizontal position. For horizontal motion at constant height: The backpack moves horizontally while maintaining constant height h, so gravitational PE = mgh remains constant (same m, same g, same h). Gravity pulls straight downward while motion is horizontal (perpendicular directions), so the displacement has no component in the gravity direction. Work by gravity = force × displacement in force direction = mg × 0 = 0 (no vertical displacement means no work by gravity). Since no work is done by or against gravity, there's no change in gravitational PE—it stays constant throughout the horizontal motion. Choice C is correct because it accurately states PE stays the same due to constant height, with motion perpendicular to gravity. Choice A incorrectly claims PE increases just because there's motion; Choice B incorrectly states PE decreases when no work is done by gravity; Choice D gives the right answer but wrong reasoning (gravity isn't zero, it just does no work on horizontal motion). The force-PE relationship clarifies: only the vertical component of motion matters for gravitational PE changes—horizontal motion at constant height involves no work by/against gravity and no PE change.
15. A student pushes a box $3\,\text{m}$ across a level floor at constant height. The motion is straight and horizontal. How does the box's gravitational potential energy (PE) change during this motion, and why?
    Answer: It stays the same because the height does not change, so $mgh$ is constant. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship for gravitational PE is PE = mgh, where h is the vertical height above a reference level—PE only changes when height changes. For horizontal motion on a level floor: the box moves 3 m horizontally with no vertical displacement (Δh = 0), gravitational force points downward (perpendicular to horizontal motion), so gravity does no work: W = F·d·cos(90°) = 0 (force perpendicular to displacement), and since height doesn't change, gravitational PE = mgh remains constant throughout the motion. The key insight is that gravitational PE depends only on vertical position (height), not horizontal position. Choice C is correct because it properly identifies that PE stays the same because height does not change, making mgh constant—this is the fundamental reason. Choice A incorrectly claims PE increases because of motion in gravity's direction, but horizontal motion is perpendicular to gravity, not in its direction; Choice B wrongly states PE decreases because gravity does work, but gravity does zero work on horizontal motion; Choice D makes the nonsensical claim that mass becomes smaller as the box moves. The force-PE relationship shows that potential energy changes require motion along the force direction: since gravity points vertically and the motion is horizontal, there's no component of motion along the gravitational force, resulting in no work done and no PE change.
16. A ball rests at the bottom of a smooth bowl. If the ball is pushed slightly up one side and released, it rolls back down. Which statement best explains why the bottom of the bowl is a stable equilibrium position?
    Answer: The bottom is a minimum in gravitational PE, so the force direction from nearby positions is toward the bottom (lower PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship for stable equilibrium is that it occurs at a potential energy minimum, where forces from all nearby positions point back toward the equilibrium position. For a ball in a bowl: the bottom is the lowest point, so it has minimum gravitational PE (PE = mgh with h smallest at bottom); from any position on the bowl's sides, gravity has a component pointing down the slope toward the bottom; this means displaced balls experience a restoring force toward the bottom (the PE minimum), causing them to roll back. The bottom is stable because it's a PE minimum with forces pointing toward it from all directions. Choice B is correct because it properly identifies the bottom as a minimum in gravitational PE and correctly explains that forces from nearby positions point toward the bottom (lower PE)—this is the defining characteristic of stable equilibrium. Choice A incorrectly calls the bottom a PE maximum when it's clearly the minimum (lowest height); Choice C wrongly claims gravity points upward at the bottom when gravity always points downward; Choice D incorrectly states PE doesn't depend on position. The force-PE relationship shows stable equilibrium occurs at PE minima: any small displacement results in a force pointing back toward the minimum, creating a restoring effect that returns the system to equilibrium, like a ball rolling back to the bowl's bottom or a pendulum returning to vertical.
17. A student compresses a spring slowly from its relaxed length to a compressed position and then lets it go. Which statement best connects work and elastic potential energy during the compression?
    Answer: The student does work against the spring force, increasing the spring's elastic potential energy. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph); for springs, when you move opposite to the force (compressing against the outward spring force), you do work against the force and potential energy increases (your work is stored as PE = ½kx²). For spring force and elastic PE during compression: the spring force points outward (toward relaxed), so compressing further means moving against this force, requiring the student to do positive work that increases elastic PE (from relaxed x=0, PE=0 to compressed x>0, PE=½kx²>0). Choice A is correct because it accurately explains that the student does work against the spring force, increasing the spring's elastic potential energy. Choice D is wrong because it states elastic potential energy decreases during compression, when compressing against the force actually increases PE (stores more energy). The force-PE relationship helps predict motion and understand energy: compressing a spring stores PE, which can later be released as KE when the spring expands. Real examples: pushing a spring-loaded toy requires work against the spring force, increasing PE, similar to how a bow stores elastic PE when drawn back.
18. Two objects have the same positive charge, so they repel each other. Consider two separations: Position 1 where they are $2\,\text{cm}$ apart, and Position 2 where they are $10\,\text{cm}$ apart. The electric force on each charge points away from the other charge. Which statement best describes the electric potential energy (PE) change when the charges move from $2\,\text{cm}$ to $10\,\text{cm}$?
    Answer: Electric PE decreases because the charges move in the direction of the force (toward lower PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases (toward lower PE, "downhill" on a PE vs position graph): for repelling charges, the electric force pushes them apart because PE is lower when separated than when close together. For electric potential energy: Two positive charges repel each other (electric force pushes them apart), and the PE relationship shows: close together (2 cm apart), high electric PE (repelling charges forced close have energy stored—like compressed spring), far apart (10 cm apart), low electric PE (separated naturally is lower energy for repulsion), and force points from high to low PE (force pushes apart, from close/high-PE toward far/low-PE), so moving from 2 cm to 10 cm (in force direction) decreases PE as the force does positive work. Choice B is correct because it accurately explains that moving in the force direction decreases electric PE (toward lower PE). Choice A wrongly states PE increases when moving in force direction (actually decreases), Choice C claims PE stays the same (but separation changes PE), and Choice D confuses by saying PE decreases only with outside work against force (but here motion is with the force). The force-PE relationship helps predict motion and understand energy: repelling charges accelerate apart when released, converting PE to KE. Systematic approach: identify force (repulsive, apart), determine PE (higher when closer), recognize force points toward lower PE, and predict PE decreases when moving in force direction.
19. A book is lifted straight up from a shelf at $1\,\text{m}$ to a higher shelf at $3\,\text{m}$. The book's weight (gravitational force) points downward. Which statement best describes how the book's gravitational potential energy changes and why?​
    Answer: The gravitational potential energy increases because the book is moved opposite the direction of gravity, so work is done against gravity and stored as potential energy. This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that forces point in the direction where potential energy decreases: gravitational force (weight) points downward because gravitational PE is lower at ground level than at height, so gravity points toward the minimum PE position. When you move an object opposite to the force direction (lifting upward against downward gravity), you do work against the force and potential energy increases (your work is stored as PE = mgh). For this scenario: the book starts at 1 m height with some initial PE = mg(1), moves to 3 m height with final PE = mg(3), and the change is ΔPE = mg(3) - mg(1) = mg(2) > 0 (PE increases). Since gravity points downward and the book moves upward (opposite to gravity), work must be done against the gravitational force, and this work equals the PE increase: W = ΔPE = mg(2). Choice A is correct because it correctly identifies that moving opposite to gravity's direction (upward motion against downward force) requires work against gravity, and this work is stored as increased gravitational potential energy. Choice B incorrectly claims upward motion lowers PE when upward motion (against gravity) increases PE; Choice C incorrectly states PE stays the same when height change always changes gravitational PE (PE = mgh depends on h); Choice D correctly notes PE increases but incorrectly claims gravity does positive work—gravity does negative work when motion is opposite to force direction (lifting up against down force). The systematic approach confirms: gravity points down toward lower PE, lifting moves up against gravity, work against force increases PE, and the work input equals the PE gain.
20. A pendulum bob swings without air resistance. Consider three locations: (1) the left highest point, (2) the lowest point, and (3) the right highest point. At which location is gravitational potential energy (PE) minimum, and how is that related to the direction of the gravitational force?
    Answer: PE is minimum at the lowest point because gravity points downward toward lower height (lower PE). This question tests understanding of the relationship between forces and potential energy—specifically, that forces point toward positions of lower potential energy and work done by or against forces changes PE. The fundamental relationship is that gravitational force always points downward (toward Earth), and this corresponds to pointing toward positions of lower gravitational PE. For a pendulum: at the highest points (left and right), height is maximum so PE = mgh is maximum; at the lowest point, height is minimum so PE = mgh is minimum (taking bottom as h=0 gives PE=0); gravity always points downward throughout the swing, which is toward the bottom position from any other position on the path. The pendulum naturally swings toward the bottom because that's where PE is minimum and gravity points toward it. Choice B is correct because it accurately identifies PE is minimum at the lowest point and correctly connects this to gravity pointing downward toward lower height (lower PE)—this shows proper understanding of the force-PE relationship. Choice A incorrectly claims PE is minimum at highest points when it's actually maximum there; Choice C wrongly states PE is same at all points when height clearly varies; Choice D correctly identifies highest points but wrongly claims they have minimum PE and incorrectly relates this to speed rather than height. The force-PE relationship explains pendulum motion: gravity always points toward the bottom (PE minimum), causing the pendulum to accelerate toward bottom from either side, converting PE to KE on the way down and KE back to PE on the way up.

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