Model Potential Energy - Middle School Physical Science

1. A pendulum bob has mass 2 kg. The lowest point of the swing is chosen as the reference level ($h=0$ m). At each endpoint of the swing, the bob is 0.5 m above the lowest point. Using $g\approx 10\,\text{m/s}^2$, what is the bob's gravitational potential energy at an endpoint (relative to the lowest point)?
   Answer: 10 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy is calculated using PE = mgh: mass m = 2 kg, gravitational field strength g = 10 m/s² (on Earth), height h = 0.5 m above the lowest point (reference level), so PE = (2 kg)(10 m/s²)(0.5 m) = 10 J. Choice D is correct because it accurately calculates PE = mgh = (2)(10)(0.5) = 10 J with proper substitution. Choice A is wrong because it claims 0 J, perhaps confusing the endpoint with the lowest point where PE=0, but at endpoint h=0.5 m >0 so PE>0. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).
2. A 1 kg book is placed on different shelves. The floor is the reference level where $h=0$ m and $PE=0$ J. Use $g\approx 10\,\text{m/s}^2$. What is the book's gravitational potential energy when it is on the high shelf at $h=2$ m?
   Answer: 20 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy is calculated using PE = mgh: mass m = 1 kg, gravitational field strength g = 10 m/s² (on Earth), height h = 2 m above the floor (reference level), so PE = (1 kg)(10 m/s²)(2 m) = 20 J. Choice B is correct because it accurately calculates PE = mgh = (1)(10)(2) = 20 J with proper substitution. Choice A is wrong because it makes a calculation error: omits g and just does mh = (1)(2) = 2 J, forgetting the gravitational field strength factor. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).
3. A $2\text{ kg}$ object is lifted from $1\text{ m}$ above the ground to $4\text{ m}$ above the ground. Using $g \approx 10\text{ m/s}^2$, what is the change in gravitational potential energy ($\Delta PE$)?
   Answer: $60\text{ J}$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). To find the change in gravitational potential energy, calculate PE at each position: initially at h₁ = 1 m, PE₁ = (2 kg)(10 m/s²)(1 m) = 20 J; finally at h₂ = 4 m, PE₂ = (2 kg)(10 m/s²)(4 m) = 80 J; therefore, change in PE = ΔPE = PE₂ - PE₁ = 80 J - 20 J = 60 J. This 60 J increase represents the work done lifting the object through a height difference of 3 m (from 1 m to 4 m)—you applied an upward force equal to the object's weight (mg = 20 N) through a distance of 3 m, doing work = force × distance = 20 N × 3 m = 60 J. Choice B is correct because it accurately calculates the change in PE as 60 J when moving from 1 m to 4 m height. Choice A (6 J) appears to have made an arithmetic error or used wrong values; Choice C (80 J) gives the final PE but not the change (forgot to subtract initial PE of 20 J); Choice D (30 J) might have calculated for only part of the height change or made calculation error. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting a 2 kg object from 1 m to 4 m requires 60 J of work—this could be done by a person, motor, or crane, and that 60 J is now stored as gravitational PE; if the object falls back to 1 m, it releases that 60 J (converting to kinetic energy during the fall).
4. A pendulum bob (mass is not needed) is at its lowest point, and that lowest point is chosen as the reference level ($h=0$ m). The bob rises to an endpoint that is 0.5 m above the lowest point. Compared to the lowest point, the bob's gravitational potential energy at the endpoint is:
   Answer: Greater This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). Comparing positions: at lowest point h = 0 m, PE = m g (0) = 0 J, at endpoint h = 0.5 m, PE = m g (0.5) = 0.5 m g J, which is greater than at lowest point since higher h means more PE (mass cancels out in comparison). Choice A is correct because it accurately compares PE values showing higher position has greater PE. Choice B is wrong because it reverses the comparison: claims less PE at higher point, when PE = mgh shows higher h gives more PE. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction). Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
5. A 2 kg backpack is lifted from the floor (reference level, $h=0$ m) to a hook that is 3 m above the floor. Using $g \approx 10\ \text{m/s}^2$, how much gravitational potential energy does the backpack gain?
   Answer: 60 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy gained is calculated using ΔPE = mgΔh: mass m = 2 kg, g = 10 m/s², change in height Δh = 3 m (from 0 m to 3 m), so ΔPE = (2)(10)(3) = 60 J. Choice C is correct because it accurately calculates the gain in PE = mgΔh = (2)(10)(3) = 60 J with proper substitution. Choice A is wrong because it makes a calculation error: arithmetic mistake like (2)(10)(0.25) or using wrong height, resulting in 5 J, when actually it's 60 J for 3 m. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
6. Two identical 2 kg backpacks are compared using $g \approx 10\,\text{m/s}^2$ and the ground as $h=0$ m. Backpack A is 1 m above the ground, and Backpack B is 3 m above the ground. How does the gravitational potential energy of Backpack B compare to Backpack A?
   Answer: Backpack B has 3 times the potential energy of Backpack A. This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). Comparing identical 2 kg backpacks: Backpack A at h = 1 m has PE = (2)(10)(1) = 20 J, Backpack B at h = 3 m has PE = (2)(10)(3) = 60 J, so B has 60 / 20 = 3 times the PE of A, since heights are in ratio 3:1 and PE ∝ h. Choice D is correct because it accurately compares PE values showing triple height means triple PE. Choice A is wrong because it claims inverse relationship: one-third, but actually PE ∝ h is direct (triple h → triple PE). Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction). Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
7. The same $1\text{ kg}$ book is first on the floor ($0\text{ m}$) and then moved to a low shelf ($1\text{ m}$) above the floor. Using $g \approx 10\text{ m/s}^2$, how much does the book's gravitational potential energy increase?
   Answer: $10\text{ J}$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). To find the increase in gravitational potential energy, calculate PE at each position and find the difference: initially on floor (h = 0 m), PE₁ = (1 kg)(10 m/s²)(0 m) = 0 J; on low shelf (h = 1 m), PE₂ = (1 kg)(10 m/s²)(1 m) = 10 J; therefore, increase in PE = PE₂ - PE₁ = 10 J - 0 J = 10 J. This 10 J increase represents the work you did lifting the book from floor to shelf—you applied an upward force equal to the book's weight (mg = 10 N) through a distance of 1 m, doing work = force × distance = 10 N × 1 m = 10 J, which is now stored as gravitational potential energy. Choice C is correct because it accurately calculates the change in PE as 10 J when moving from 0 m to 1 m height. Choice A (1 J) appears to have omitted g, calculating just m×Δh = 1×1 = 1 J; Choice B (0 J) incorrectly suggests no change in PE despite changing height; Choice D (100 J) makes an arithmetic error, possibly calculating 1×10×10 instead of 1×10×1. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 1 m shelf increases its PE by (1)(10)(1) = 10 J—you did 10 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 10 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).
8. A 2 kg object is lifted from $h=1$ m to $h=4$ m above the ground (ground is the reference level). Use $g\approx 10\,\text{m/s}^2$. How much does its gravitational potential energy increase?
   Answer: 60 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The increase in gravitational potential energy is calculated as ΔPE = mg Δh: mass m = 2 kg, g = 10 m/s², Δh = 4 m - 1 m = 3 m, so ΔPE = (2)(10)(3) = 60 J (alternatively, PE at 1 m = 20 J, at 4 m = 80 J, increase = 60 J). Choice A is correct because it accurately calculates the increase in PE as mg Δh = (2)(10)(3) = 60 J. Choice B is wrong because it calculates PE at 4 m = 80 J but forgets to subtract the initial PE, giving the final PE instead of the increase. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
9. Two identical flower pots are placed at the same height of 2 m above the ground (ground is the reference level). Pot A has mass 2 kg and Pot B has mass 4 kg. Using $g \approx 10\ \text{m/s}^2$, how does Pot B's gravitational potential energy compare to Pot A's?
   Answer: Pot B has twice as much gravitational potential energy as Pot A. This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). At the same height h = 2 m, comparing different masses: 2 kg pot has PE = (2)(10)(2) = 40 J, 4 kg pot has PE = (4)(10)(2) = 80 J (exactly double)—gravitational potential energy is directly proportional to mass at constant height: PE ∝ m. Choice C is correct because it properly identifies the proportional relationship: double mass → double PE. Choice D is wrong because it applies wrong relationship: claims doubling mass quadruples PE (confusing with KE-speed squared), when actually PE ∝ m is linear (double m → double PE, not quadruple). Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
10. A roller coaster cart has a mass of $500\text{ kg}$. At the top of a hill it is $20\text{ m}$ above the ground (reference level). Using $g \approx 10\text{ m/s}^2$, what is its gravitational potential energy at the top?
    Answer: $100{,}000\text{ J}$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy is calculated using PE = mgh: mass m = 500 kg, gravitational field strength g = 10 m/s² (on Earth), height h = 20 m above the ground (reference level), so PE = (500 kg)(10 m/s²)(20 m) = 100,000 J. This means the roller coaster cart has 100,000 Joules of gravitational potential energy at the top of the hill—if it were to roll down from 20 m height to ground level, this 100,000 J of PE would convert to kinetic energy (in ideal case with no friction, KE at ground would be 100,000 J, giving the cart significant speed). Choice C is correct because it correctly calculates PE = mgh = (500)(10)(20) = 100,000 J with proper substitution. Choice A (10,000 J) appears to have made an arithmetic error, possibly calculating 500×20 = 10,000 without including g; Choice B (50,000 J) might have used wrong height (10 m instead of 20 m) giving 500×10×10 = 50,000; Choice D (1,000 J) severely underestimates, perhaps dividing instead of multiplying or using wrong values. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: a 500 kg roller coaster cart at 20 m height has PE = 100,000 J—this enormous energy was provided by the lift mechanism pulling the cart up the first hill, and as the cart descends, this PE converts to KE making the ride thrilling; water behind a dam at similar heights stores millions of joules as gravitational PE, which converts to flowing water's KE to spin turbines for hydroelectric power.
11. A 5 kg bucket is lifted from $h=1$ m above the ground to $h=4$ m above the ground. (The ground is the reference level.) Using $g \approx 10\ \text{m/s}^2$, what is the change in gravitational potential energy, $\Delta PE$, of the bucket?
    Answer: 150 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The change in gravitational potential energy is calculated using ΔPE = mgΔh: mass m = 5 kg, g = 10 m/s², change in height Δh = 3 m (from 1 m to 4 m), so ΔPE = (5)(10)(3) = 150 J; alternatively, initial PE = (5)(10)(1) = 50 J, final PE = (5)(10)(4) = 200 J, ΔPE = 200 - 50 = 150 J. Choice C is correct because it accurately calculates ΔPE = mgΔh = (5)(10)(3) = 150 J with proper substitution. Choice D is wrong because it makes a calculation error: uses final PE = 200 J instead of the change ΔPE = 150 J, confusing total PE with the change. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
12. A 4 kg toolbox is on a platform 2 m above the ground (ground is the reference level). If the toolbox is moved to a platform 4 m above the ground, how does its gravitational potential energy change? Assume $g \approx 10\ \text{m/s}^2$.
    Answer: It doubles. This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). Comparing the same 4 kg object at different heights: at h = 2 m, PE = (4)(10)(2) = 80 J, and at h = 4 m, PE = (4)(10)(4) = 160 J—the PE doubled (80 → 160 J) when height doubled (2 → 4 m), demonstrating that gravitational potential energy is directly proportional to height: PE ∝ h. Choice A is correct because it properly identifies the proportional relationship: double height → double PE. Choice C is wrong because it applies wrong relationship: claims doubling height quadruples PE (confusing with KE-speed squared), when actually PE ∝ h is linear (double h → double PE, not quadruple). Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).
13. A 5 kg object has gravitational potential energy of 150 J relative to the ground. Using $g\approx 10\,\text{m/s}^2$, how high above the ground is the object?
    Answer: 3 m This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). To find height from PE, rearrange h = PE / (mg): PE = 150 J, m = 5 kg, g = 10 m/s², so h = 150 / (5 × 10) = 150 / 50 = 3 m above the ground. Choice B is correct because it properly applies the rearranged formula to determine h = PE / (mg) = 150 / 50 = 3 m. Choice A is wrong because it makes a calculation error: perhaps h = 150 / (5 × 2) = 15 m or using wrong g. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
14. A 3 kg object has gravitational potential energy of 150 J relative to the ground. Using $g \approx 10\,\text{m/s}^2$, how high above the ground is the object?
    Answer: 5 m This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). To find height, rearrange PE = mgh to h = PE / (mg): with PE = 150 J, m = 3 kg, g = 10 m/s², h = 150 / (3×10) = 150 / 30 = 5 m. This means the object is 5 m above the ground, storing 150 J of PE that could convert to KE if it falls. Choice C is correct because it correctly calculates h = PE / (mg) = 150 / (3×10) = 5 m. Choice D is wrong because it makes a calculation error: perhaps omitting m and calculating 150 / 10 = 15 m, but mass must be included since PE ∝ m. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction). Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
15. A student uses the formula $PE=mgh$ to find gravitational potential energy near Earth's surface. Which set of quantities is needed to calculate $PE$ for an object at rest?
    Answer: Mass, height above the reference point, and $g$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). For an object at rest, PE depends only on m, g, and h—no speed or other factors, as kinetic energy would involve speed but PE is positional. Choice A is correct because it properly identifies the quantities in PE = mgh: mass, height above reference, and g. Choice B is wrong because it includes speed, which is for kinetic energy (KE = ½mv²), not gravitational PE. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction). Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
16. A $3\text{ kg}$ backpack is on a ledge $2\text{ m}$ above the ground. Using $g \approx 10\text{ m/s}^2$, which expression correctly gives its gravitational potential energy?
    Answer: $PE = mgh = (3)(10)(2)$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). For the 3 kg backpack at 2 m height, the correct expression for gravitational potential energy is PE = mgh = (3 kg)(10 m/s²)(2 m) = 60 J, which shows proper application of the formula with all three factors: mass (3 kg), gravitational field strength (10 m/s²), and height (2 m). This expression correctly represents that PE depends on all three factors multiplied together—doubling any one factor doubles the PE. Choice A is correct because it properly applies the formula PE = mgh with correct substitution of values (3)(10)(2). Choice B shows the kinetic energy formula (½mv²) which is wrong type of energy—KE depends on velocity, not height; Choice C incorrectly divides by height (mg/h) which would give wrong units and makes no physical sense; Choice D omits gravitational field strength g, using just mh which would give wrong units (kg·m instead of Joules) and wrong value. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: a 3 kg backpack on a 2 m ledge has PE = (3)(10)(2) = 60 J—someone did 60 J of work carrying it up there, and if it falls, that 60 J converts to kinetic energy; understanding the correct formula helps calculate energy storage in elevated objects from water towers to ski lifts.
17. A $1\text{ kg}$ book is placed on a shelf that is $2\text{ m}$ above the floor. Take $g \approx 10\text{ m/s}^2$ and the floor as the reference level ($h=0$). What is the book's gravitational potential energy?
    Answer: $20\text{ J}$ This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy is calculated using PE = mgh: mass m = 1 kg, gravitational field strength g = 10 m/s² (on Earth), height h = 2 m above the floor (reference level), so PE = (1 kg)(10 m/s²)(2 m) = 20 J. This means the book has 20 Joules of gravitational potential energy at this position—if it were to fall from 2 m height to the floor, this 20 J of PE would convert to kinetic energy (in ideal case with no friction, KE at floor would be 20 J). Choice B is correct because it correctly calculates PE = mgh = (1)(10)(2) = 20 J with proper substitution. Choice A (5 J) makes calculation error: possibly used wrong formula or arithmetic mistake; Choice C (200 J) likely multiplied incorrectly (perhaps 1×10×20 or 10×20); Choice D (2 J) appears to have omitted the gravitational field strength g, calculating just m×h = 1×2 = 2 instead of m×g×h. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).
18. A 4 kg object is raised from 2 m to 6 m above the ground. If $g\approx 10\,\text{m/s}^2$, how does its gravitational potential energy at 6 m compare to its gravitational potential energy at 2 m?
    Answer: It is 3 times as large This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). Comparing the same 4 kg object at different heights: at h = 2 m, PE = (4)(10)(2) = 80 J, and at h = 6 m, PE = (4)(10)(6) = 240 J—the PE is 240/80 = 3 times as large when height is 6/2 = 3 times as large, demonstrating that gravitational potential energy is directly proportional to height: PE ∝ h. Choice A is correct because it properly identifies the proportional relationship: triple height (2 → 6 m) → triple PE. Choice B is wrong because it applies wrong relationship: claims 4 times, perhaps confusing with something else, but actually PE ∝ h is linear (triple h → triple PE, not quadruple). Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
19. Three objects sit on a table that is 1.5 m above the ground (ground is the reference level). Their masses are 1 kg, 2 kg, and 5 kg. Using $g \approx 10\,\text{m/s}^2$, which object has the greatest gravitational potential energy?
    Answer: The 5 kg object This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). At the same height h = 1.5 m, comparing different masses: 1 kg object has PE = (1)(10)(1.5) = 15 J, 2 kg object has PE = (2)(10)(1.5) = 30 J, and 5 kg object has PE = (5)(10)(1.5) = 75 J—the 5 kg has the greatest, showing gravitational potential energy is directly proportional to mass at constant height: PE ∝ m. The heavier object has more PE at the same height because it required more work to lift it there (lifting 5 kg through 1.5 m takes more work than lifting 1 kg through same 1.5 m), and it stores that work as gravitational potential energy. Choice C is correct because it properly identifies the proportional relationship: greater mass → greater PE at same height. Choice D is wrong because it misunderstands proportionality: claims all have same PE since same height, but actually PE ∝ m, so different masses have different PE even at same h. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction). Water behind dam at height h has gravitational PE = mgh (enormous for millions of kg of water at tens of meters height), which converts to KE as water flows down through turbines (hydroelectric power: stored PE → flowing KE → rotational KE of turbine → electrical energy), showing how gravitational PE is useful stored energy we can harness.
20. A 1 kg book is placed on three different levels in a classroom. The floor is the reference level where $h=0$ m. Shelf heights above the floor are 1 m (low shelf) and 2 m (high shelf). Using $g \approx 10\ \text{m/s}^2$, what is the book's gravitational potential energy on the 2 m shelf?
    Answer: 20 J This question tests understanding of gravitational potential energy calculated using PE = mgh, where energy depends on mass, height, and gravitational field strength. Gravitational potential energy (PE = mgh) is the energy an object has due to its position in a gravitational field—specifically, its height above some chosen reference level: an object at height h has the potential to fall through that height, converting potential energy to kinetic energy, and the amount of PE represents how much work was done to lift the object to that height (or how much work it could do falling back down). The formula shows three factors: m is the object's mass in kilograms (more mass → more PE), g is Earth's gravitational field strength ≈ 10 m/s² or 10 N/kg (on Earth's surface, essentially constant), and h is height in meters above the chosen reference level where PE = 0 (ground, floor, table—you choose the reference). The gravitational potential energy is calculated using PE = mgh: mass m = 1 kg, gravitational field strength g = 10 m/s² (on Earth), height h = 2 m above the floor (reference level), so PE = (1 kg)(10 m/s²)(2 m) = 20 J. Choice C is correct because it accurately calculates PE = mgh = (1)(10)(2) = 20 J with proper substitution. Choice A is wrong because it makes a calculation error: uses wrong formula or arithmetic mistake, like omitting g and calculating just mh = (1)(2) = 2 J, when actually PE includes g. Working with gravitational potential energy: (1) choose reference level where PE = 0 (usually ground or lowest point in scenario), (2) measure all heights from this reference (height above = positive h, below would be negative h), (3) calculate PE = mgh for each position using mass in kg, g = 10 m/s², height in m, (4) compare: higher positions have more PE (proportional to h), heavier objects have more PE at same height (proportional to m), and (5) understand meaning: PE is energy stored by position, represents work done lifting object, can convert to KE if object falls. Practical examples: lifting 1 kg book from floor to 2 m shelf increases its PE by (1)(10)(2) = 20 J—you did 20 J of work lifting it, and that energy is now stored as PE; if book falls back to floor, that 20 J converts to KE as it falls (speeds up gaining kinetic energy while losing potential energy, total mechanical energy conserved if no friction).

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