Compare Potential Energy - Middle School Physical Science

1. Which has greater elastic potential energy: same $x$, spring constant $k$ versus $2k$?

   $2k$ (it is $2$ times larger). Energy is linear in $k$: double $k$ doubles energy.

2. State the formula for elastic potential energy stored in an ideal spring.

   $U_s = \frac{1}{2}kx^2$. Where $k$ is spring constant and $x$ is displacement.

3. Which arrangement has greater gravitational potential energy: mass $m$ at $h_1$ or at $h_2$ if $h_2 > h_1$?

   The mass at $h_2$ has greater $U_g$. Higher position means more gravitational PE since $U_g = mgh$.

4. If a spring's stretch changes from $x$ to $2x$, by what factor does elastic potential energy change?

   It increases by a factor of $4$. Since $U_s propto x^2$, doubling $x$ gives $(2x)^2 = 4x^2$.

5. What is the key comparison rule for gravitational potential energy when $g$ is the same for all cases?

   Compare $m\times h$ (larger $m h$ means larger $U_g$). Since $U_g = mgh$ and $g$ is constant.

6. Which arrangement has greater $U_g$: same $m$, A at $0\,\text{m}$ or B at $-2\,\text{m}$ relative to reference?

   A at $0\,\text{m}$. Higher position (0 m > -2 m) means greater gravitational potential energy.

7. Which has greater $PE_e$: $k=100\,\text{N/m}, x=0.20\,\text{m}$ or $k=200\,\text{N/m}, x=0.10\,\text{m}$?

   The $k=100, x=0.20$ case (larger $kx^2$). First: $PE_e = 4$ J; second: $PE_e = 1$ J.

8. Which spring stores more $PE_e$ for the same stretch $x$: larger $k$ or smaller $k$?

   Larger $k$ stores more $PE_e$. $PE_e$ increases linearly with spring constant $k$.

9. What happens to $PE_g$ when an object is lifted at constant speed by height $\Delta h$?

   $PE_g$ increases by $\Delta PE_g = mg\Delta h$. Work done against gravity equals the change in gravitational PE.

10. What is gravitational potential energy near Earth in terms of mass, gravity, and height?

    $U_g = mgh$. Formula shows potential energy depends on mass, gravity, and height.

11. Two books have the same mass; one is at height $2h$ and one at $h$. Which has greater $U_g$?

    The book at $2h$ has greater $U_g$. Double the height doubles the gravitational potential energy.

12. What happens to gravitational potential energy when an object is lifted higher above the same reference level?

    It increases. Higher $h$ means larger $mgh$, thus more potential energy.

13. Which object has greater $PE_g$: $2\,\text{kg}$ at $3\,\text{m}$ or $3\,\text{kg}$ at $2\,\text{m}$ (same $g$)?

    They are equal ($m \times h = 6$ in both cases). Since $PE_g = mgh$, both give $PE_g = 6g$ joules.

14. Which statement is correct about comparing $U_g$ values between two objects?

    Only differences in $U_g$ are physically meaningful. Absolute values depend on reference choice; only changes matter physically.

15. What is the key condition for a spring to have elastic potential energy in a system?

    It has $U_s$ when it is stretched or compressed from equilibrium. Displacement from equilibrium stores elastic energy in the spring.

16. Identify which spring stores more energy at the same stretch $x$: spring with $k=200$ or $k=100$ (SI units).

    $k=200$ stores twice as much $U_s$. $U_s = \frac{1}{2}kx^2$, so doubling $k$ doubles energy.

17. If height is doubled while mass stays the same, how does $U_g$ change?

    $U_g$ doubles. Since $U_g = mgh$, doubling $h$ doubles the energy.

18. Which has greater gravitational potential energy: $3\text{ kg}$ at $4\text{ m}$ or $6\text{ kg}$ at $2\text{ m}$?

    They are equal. Both equal $24\text{ J}$ using $U_g = mgh$ with $g = 10\text{ m/s}^2$.

19. Which has greater gravitational potential energy: mass $2m$ at height $h$ or mass $m$ at $h$?

    Mass $2m$ at $h$ has greater $U_g$. Doubling mass doubles gravitational potential energy.

20. Which has greater $U_g$ increase: raising mass $m$ by $2\text{ m}$ or raising mass $2m$ by $1\text{ m}$ (same $g$)?

    They are equal (both give $\Delta U_g = 2m g$). Both give $\Delta U_g = 2mg$ since $(m)(2g) = (2m)(g)$.

21. What type of potential energy depends mainly on an object's height in a gravitational field?

    Gravitational potential energy. Height determines energy stored in Earth's gravitational field.

22. Which has greater $U_g$: $m$ at height $h$ or $2m$ at height $\frac{1}{2}h$ (same $g$)?

    They are equal. Both have $mh = m cdot h = 2m cdot \frac{h}{2} = mh$.

23. What is the key condition for an object to have gravitational potential energy in a system?

    It has $U_g$ when it is at a height in a gravitational field. Height above a reference point creates gravitational potential energy.

24. Which statement correctly compares potential energy in different arrangements of the same system?

    The arrangement with larger $U$ has more stored energy. Higher potential energy means more energy available to do work.

25. What type of potential energy is stored when a spring is compressed or stretched?

    Elastic potential energy. Energy stored when elastic materials deform from equilibrium.

26. Which stores more elastic potential energy: spring A ($k$, $x$) or B ($2k$, $\frac{1}{2}x$)?

    Spring A. A: $\frac{1}{2}kx^2$; B: $\frac{1}{2}(2k)(\frac{x}{2})^2 = \frac{1}{4}kx^2$.

27. What is the change in gravitational potential energy when lifting a mass $m$ by a height change $\Delta h$?

    $\Delta U_g = mg\Delta h$. Derived from $U_g = mgh$ for a change in height.

28. Which object has greater gravitational potential energy: same $m$, heights $h_1 > h_2$?

    The object at height $h_1$. Higher position means greater $mgh$ value.

29. What is the sign of $PE_g$ change when an object moves downward by distance $\Delta h$ (with $\Delta h<0$)?

    $\Delta PE_g$ is negative. Moving down decreases height, so $mgh$ decreases.

30. Which arrangement has more $U_s$: $k=100\,\text{N/m}$, $x=0.10\,\text{m}$ or $k=50\,\text{N/m}$, $x=0.10\,\text{m}$?

    $k=100\,\text{N/m}$ case has more $U_s$. Stiffer spring ($k=100$) stores twice the energy at same stretch.

31. Which object has greater gravitational potential energy if both are at the same height: larger mass or smaller mass?

    The larger mass has greater $U_g$. Since $U_g = mgh$, larger $m$ means larger potential energy.

32. Which has greater elastic potential energy: $k=100\text{ N/m}, x=0.20\text{ m}$ or $k=50\text{ N/m}, x=0.20\text{ m}$?

    $k=100\text{ N/m}$, $x=0.20\text{ m}$. Larger $k$ means more PE since $U_s = \frac{1}{2}kx^2$.

33. Which arrangement stores more elastic potential energy: spring compressed by $x$ or stretched by $x$ (same $k$)?

    They store the same $U_s$. Since $U_s$ depends on $x^2$, compression and stretch of same magnitude give same energy.

34. If spring displacement doubles from $x$ to $2x$ (same $k$), how does $U_s$ change?

    $U_s$ increases by a factor of $4$. Since $U_s \propto x^2$, doubling $x$ quadruples $U_s$.

35. Which arrangement has more elastic potential energy: a spring compressed by $x$ or stretched by $x$ (same $k$)?

    They are equal ($U_s$ depends on $x^2$). Compression and stretch by same amount store equal energy.

36. Which system has greater total potential energy: same $m$, same $h$, plus a stretched spring ($x>0$) vs no spring stretch ($x=0$)?

    The system with the stretched spring ($x>0$). Stretched spring adds elastic potential energy to the gravitational energy.

37. If mass triples from $m$ to $3m$ at the same height, how does $U_g$ change?

    $U_g$ triples. PE is proportional to mass, so tripling $m$ triples $U_g$.

38. Which has greater elastic potential energy: $k=100\text{ N/m}, x=0.10\text{ m}$ or $k=100\text{ N/m}, x=0.20\text{ m}$?

    $x=0.20\text{ m}$. Larger stretch means more PE since $U_s \propto x^2$.

39. What formula gives gravitational potential energy near Earth for mass $m$ at height $h$?

    $U_g = mgh$. Mass times gravitational acceleration times height gives gravitational potential energy.

40. Which arrangement has greater $PE_g$: mass $m$ at $h=5\text{ m}$ or mass $2m$ at $h=2\text{ m}$ (same $g$)?

    Mass $m$ at $5\text{ m}$ has greater $PE_g$. $m \times 5 = 5m$ while $2m \times 2 = 4m$, so first is greater.

41. Identify the arrangement with greater gravitational potential energy: same $m$, heights $h=0\,\text{m}$ vs $h=2\,\text{m}$.

    The object at $h=2\,\text{m}$. Higher position has greater $mgh$ than ground level.

42. Which has greater elastic potential energy: same $k$, stretch $x$ versus stretch $2x$?

    Stretch $2x$ (it is $4$ times larger). Energy scales with $x^2$: $(2x)^2 = 4x^2$.

43. What is the key comparison rule for spring potential energy when comparing springs using the same units?

    Compare $k\times x^2$ (larger $k x^2$ means larger $U_s$). Since $U_s = \frac{1}{2}kx^2$ for all springs.

44. Which arrangement has greater $PE_g$ relative to the same reference: object at $+2\,\text{m}$ or at $-2\,\text{m}$?

    The object at $+2\,\text{m}$ has greater $PE_g$. Positive height means above reference, so higher $PE_g$.

45. What happens to gravitational potential energy when height increases by $\Delta h$ (with $m$ constant)?

    $\Delta U_g = mg\Delta h$. Change in height multiplied by $mg$ gives change in $U_g$.

46. What is the formula for elastic potential energy of a spring with constant $k$ stretched $x$?

    $U_s = \frac{1}{2}kx^2$. Spring constant times displacement squared, divided by two.

47. What type of potential energy depends on the positions of charged particles in an electric field?

    Electric potential energy. Energy stored due to positions of charges in electric fields.

48. Which arrangement has greater $U_s$: spring compressed by $x$ or stretched by $x$ (same $k$)?

    They are equal: both have $U_s = \frac{1}{2}kx^2$. $U_s$ depends on $x^2$, so direction doesn't matter.

49. Which option best describes when gravitational potential energy is defined as $0$ at a reference level?

    At the chosen reference height (zero level). We define $U_g = 0$ at any convenient reference height.

50. Which stores more elastic potential energy: spring A ($k$, $3x$) or B ($9k$, $x$)?

    They are equal. A: $\frac{1}{2}k(3x)^2 = \frac{9}{2}kx^2$; B: $\frac{1}{2}(9k)x^2 = \frac{9}{2}kx^2$.

51. For opposite charges, does electric potential energy increase or decrease as the charges move closer together?

    It decreases as opposite charges move closer. Opposite charges attract, so separating them stores energy.

52. Identify the correct comparison: $U_g$ for $m=4\text{ kg}, h=2\text{ m}$ versus $m=2\text{ kg}, h=4\text{ m}$ (same $g$).

    They are equal ($m h$ is the same). Both give $mgh = 8g$ joules, confirming equal PE.

53. Which has greater $U_g$: same object on the Moon or on Earth at the same height $h$?

    On Earth. Earth's stronger gravity ($g_{Earth} > g_{Moon}$) gives higher $U_g$ at same $h$.

54. What happens to gravitational potential energy $U_g$ when mass triples from $m$ to $3m$ (same $h$)?

    $U_g$ triples. Since $U_g = mgh$, tripling $m$ triples $U_g$.

55. If stretch increases from $x$ to $2x$ (same $k$), how does $U_s$ change?

    $U_s$ increases by a factor of $4$. Elastic energy depends on $x^2$, so doubling $x$ quadruples energy.

56. What reference level is used to define $PE_g = 0$ in a problem?

    The chosen zero-height reference level. We measure height (and thus $PE_g$) from this chosen level.

57. Which statement is correct about comparing $U_g$ values if the same zero height is used for both cases?

    Higher position means greater $U_g$ for the same $m$. With same reference, higher objects have more $U_g$.

58. An object moves from $h=10\text{ m}$ to $h=3\text{ m}$. Is $\Delta U_g$ positive, negative, or zero?

    Negative. Height decreases, so potential energy decreases.

59. In general, when like charges are pushed closer together, does electric potential energy increase or decrease?

    Increase. Work is needed to overcome repulsion between like charges.

60. Which system arrangement stores more elastic potential energy: compressed spring by $x$ or uncompressed spring?

    Compressed by $x$ stores more $U_s$. Any deformation from equilibrium stores elastic potential energy.

61. State the formula for gravitational potential energy near Earth using mass, gravity, and height.

    $U_g = mgh$. Where $m$ is mass, $g$ is gravity acceleration, and $h$ is height.

62. Which stores more elastic potential energy: a spring stretched $3x$ or stretched $x$ (same $k$)?

    Stretched $3x$. Since $U_s propto x^2$, $(3x)^2 = 9x^2$ vs $x^2$.

63. Which arrangement has more total potential energy: $U_g=30$ J and $U_s=10$ J, or $U_g=20$ J and $U_s=25$ J?

    The second arrangement ($45$ J total) has more. Total potential energy: first has $40$ J, second has $45$ J.

64. Which object has greater $PE_g$: $m=3\text{ kg}$ at $h=2\text{ m}$ or $m=2\text{ kg}$ at $h=3\text{ m}$ (same $g$)?

    They are equal ($m\times h$ is $6$ for both). Both have $PE_g = 6mg$ since $3 \times 2 = 2 \times 3 = 6$.

65. Which arrangement has greater $U_g$: $m=2\text{ kg}$ at $h=3\text{ m}$ or $m=3\text{ kg}$ at $h=2\text{ m}$?

    They are equal ($m\cdot h$ is the same). Both give $U_g = 6mg$ (same product of $m$ and $h$).

66. If mass triples while height stays constant, by what factor does $U_g$ change?

    $U_g$ triples (factor of $3$). Since $U_g = mgh$, tripling $m$ triples $U_g$.

67. Which arrangement has greater gravitational potential energy: same mass at height $h$ or at height $2h$?

    Height $2h$ has greater $U_g$. $U_g = mgh$, so doubling height doubles potential energy.

68. Which has greater spring potential energy: stretch $x$ or $2x$ for the same spring constant $k$?

    Stretch $2x$ (energy becomes $4$ times larger). PE depends on $x^2$, so doubling stretch quadruples energy.

69. Identify which energy depends on position in a gravitational field: kinetic energy or gravitational potential energy.

    Gravitational potential energy. Only gravitational PE depends on vertical position.

70. What is elastic potential energy stored in an ideal spring in terms of $k$ and stretch $x$?

    $U_s = \frac{1}{2}kx^2$. Shows elastic energy depends on spring constant and stretch squared.

71. Which has greater $U_g$: $1\text{ kg}$ at $10\text{ m}$ or $5\text{ kg}$ at $2\text{ m}$ (same $g$)?

    They are equal. Both have $mh = 1 \times 10 = 5 \times 2 = 10$.

72. What is conserved in a closed system when only conservative forces act (gravity and springs)?

    Total mechanical energy ($K + U$) is conserved. Energy transforms between kinetic and potential but sum stays constant.

73. Which stores more elastic potential energy: $k=100\,\text{N/m}$, $x=0.20\,\text{m}$ or $k=50\,\text{N/m}$, $x=0.30\,\text{m}$?

    $k=50\,\text{N/m}$, $x=0.30\,\text{m}$ (larger $kx^2$). $U_s = \frac{1}{2}kx^2$: $50(0.09) = 4.5 > 100(0.04) = 4$ J.

74. What relationship describes gravitational potential energy near Earth's surface?

    $U_g = mgh$. Mass times gravity times height gives gravitational potential energy.

75. Which has greater gravitational potential energy relative to the same reference: an object on a shelf or on the floor?

    The object on the shelf (higher $h$). Shelf is higher, so $mgh$ is greater than floor level.

76. Identify the condition under which gravitational potential energy can be negative.

    When the chosen reference level makes $h<0$. Height below reference level gives negative $U_g$.

77. Which object has greater gravitational potential energy if both have the same mass: higher height or lower height?

    The higher height has greater $U_g$. Since $U_g = mgh$, larger $h$ means larger potential energy.

78. Identify the energy type: energy stored when a spring or rubber band is stretched or compressed.

    Elastic potential energy. Energy from deforming elastic materials.

79. Which has greater elastic potential energy: same spring stretched $2x$ or stretched $x$?

    Stretch $2x$ has greater $PE_e$. $PE_e$ depends on $x^2$, so doubling $x$ quadruples energy.

80. Identify the correct comparison: same $m$, height doubles from $h$ to $2h$; how does $U_g$ change?

    $U_g$ doubles. Since $U_g = mgh$, doubling $h$ doubles the product.

81. In general, when opposite charges are pulled farther apart, does electric potential energy increase or decrease?

    Increase. Work is needed to separate attracting charges.

82. If a spring's stretch changes from $x$ to $2x$, by what factor does $U_s$ change?

    $U_s$ increases by a factor of $4$. Since $U_s \propto x^2$, doubling stretch quadruples energy.

83. Which arrangement stores more elastic energy: spring A with $k=4\text{ N/m}$ at $x=2\text{ m}$ or spring B with $k=8\text{ N/m}$ at $x=1\text{ m}$?

    Spring A stores more $PE_e$. Spring A: $\frac{1}{2}(4)(2^2) = 8$ J; Spring B: $\frac{1}{2}(8)(1^2) = 4$ J.

84. For the same stretch $x$, which stores more elastic potential energy: spring with $2k$ or spring with $k$?

    The spring with $2k$ stores more $U_s$. $U_s$ is directly proportional to spring constant $k$.

85. Which has greater spring potential energy: spring with $k$ or with $2k$ at the same stretch $x$?

    Spring with $2k$ has greater $U_s$. Stiffer spring stores more energy at same stretch.

86. What is the elastic potential energy formula for a spring with constant $k$ stretched by $x$?

    $PE_e = \frac{1}{2}kx^2$. Formula for energy stored in a stretched or compressed spring.

87. Identify the correct comparison: same $h$, mass triples from $m$ to $3m$; how does $U_g$ change?

    $U_g$ triples. Since $U_g = mgh$, tripling $m$ triples the product.

88. Which has greater $U_g$: object A at $m$ and $2h$, or object B at $2m$ and $h$?

    They are equal. $m \cdot 2h = 2m \cdot h$, so both have same $U_g$.

89. What happens to $U_g$ if height changes from $h$ to $3h$ while $m$ stays constant?

    It becomes $3$ times larger. $U_g$ is directly proportional to height.

90. What is the elastic potential energy formula for a spring stretched or compressed by $x$?

    $U_s = \frac{1}{2}kx^2$. Where $k$ is spring constant and $x$ is displacement from equilibrium.

91. Which arrangement has greater total potential energy: a spring at $x=0$ and object at $h$ or the same spring at $x\neq 0$ and object at $h$?

    The arrangement with $x\neq 0$ has greater total potential energy. Stretched/compressed spring adds $U_s > 0$ to the system's total.

92. Which has greater $U_g$: object at $h=0\text{ m}$ or at $h=5\text{ m}$, same $m$ and same reference level?

    The object at $h=5\text{ m}$. Higher position has greater potential energy.

93. Which has greater gravitational potential energy: same $h$, masses $m_1 > m_2$?

    The object with mass $m_1$. Larger mass increases $mgh$ proportionally.

94. Which has more $U_g$: A ($m$, $3h$) or B ($3m$, $h$) near Earth?

    They are equal. Both equal $3mgh$ since $U_g$ is proportional to both mass and height.

95. Identify the correct comparison: for equal $m$ and $g$, how does $PE_g$ change as $h$ increases?

    $PE_g$ increases linearly with $h$. From $PE_g = mgh$, doubling $h$ doubles $PE_g$.

96. Which location gives greater $PE_g$ for the same $m$ and $h$: larger $g$ or smaller $g$?

    Larger $g$ gives greater $PE_g$. $PE_g = mgh$ increases linearly with $g$.

97. Which object has greater gravitational potential energy: same $m$, heights $h_1 > h_2$?

    The object at height $h_1$. Greater height means greater gravitational potential energy.

98. In a stable arrangement, is potential energy relatively high or relatively low compared with nearby arrangements?

    Relatively low potential energy. Systems naturally move toward lower potential energy states.

99. Which comparison is correct for elastic energy: if stretch doubles, $PE_e$ changes by what factor?

    It increases by a factor of $4$. Since $PE_e \propto x^2$, doubling $x$ gives $(2x)^2 = 4x^2$.

100. What is the correct relationship between work done lifting slowly and the increase in gravitational potential energy?

     Work done equals $\Delta PE_g$. Energy conservation: work input becomes stored PE.

101. Which arrangement has greater $PE_g$: same object at the top of a hill or halfway up the hill?

     At the top of the hill (greater height). Maximum height gives maximum gravitational potential energy.

102. What happens to gravitational potential energy when an object moves downward by $\Delta h$?

     $U_g$ decreases by $mg\Delta h$. Moving down reduces height, decreasing gravitational potential energy.

103. Which stores more energy: same $k$ and $|x|$, a spring stretched by $+x$ or compressed by $-x$?

     They are equal (depends on $x^2$). Energy depends on $x^2$, not direction of displacement.

104. Which spring stores more elastic potential energy: same $x$, spring constants $k_1 > k_2$?

     The spring with constant $k_1$. Stiffer springs (larger $k$) store more energy for same displacement.

105. What happens to elastic potential energy when a stretched spring returns toward its natural length?

     Elastic potential energy decreases. Less stretch means less stored elastic energy.

106. Which position has greater $U_g$: a book on a top shelf or the same book on the floor (same reference)?

     The book on the top shelf has greater $U_g$. Higher position means greater height $h$ in $U_g = mgh$.

107. Which has greater gravitational potential energy: mass $2m$ at height $h$ or mass $m$ at height $h$?

     Mass $2m$ at height $h$ has greater $PE_g$. PE is directly proportional to mass in the formula $PE_g = mgh$.

108. Which change increases spring potential energy more: doubling $x$ or doubling $k$ (other variable constant)?

     Doubling $x$ increases $U_s$ more. Since $x$ is squared in $U_s = \frac{1}{2}kx^2$, doubling $x$ quadruples $U_s$.

109. Which has greater gravitational potential energy: same $h$, masses $m_1 > m_2$?

     The object with mass $m_1$. Greater mass at same height stores more gravitational potential energy.

110. Which option increases gravitational potential energy: lifting, lowering, or moving horizontally at same height?

     Lifting increases $U_g$. Only vertical upward motion increases gravitational potential energy.

111. What is the formula for gravitational potential energy near Earth for mass $m$ at height $h$?

     $U_g = mgh$. Gravitational PE equals mass times gravity times height.

112. Which has greater gravitational potential energy: same mass at $2h$ or at $h$?

     Same mass at $2h$ has greater $U_g$. Doubling height doubles gravitational potential energy.

113. Identify which stores more potential energy: a spring stretched to $x$ or the same spring stretched to $\frac{x}{2}$.

     Stretched to $x$ stores more $PE_e$. $x^2 > (\frac{x}{2})^2 = \frac{x^2}{4}$, so full stretch stores more.

114. Which arrangement has greater elastic potential energy: same $k$, compressions $x_1$ and $x_2$ with $x_2>x_1$?

     The arrangement with $x_2$ has greater $U_s$. Greater compression/stretch means greater elastic potential energy.

115. Which arrangement has greater $U_g$: same $m$, heights $2h$ versus $h$?

     The object at height $2h$. Doubling height doubles $U_g$ since $U_g \propto h$.

116. Choose the correct comparison: same $m$ and $h$, but larger $g$ versus smaller $g$ for $U_g$.

     Larger $g$ gives larger $U_g$. Stronger gravity field increases potential energy at same height.

117. Which arrangement has more $U_g$: same mass at $h=4\text{ m}$ or at $h=6\text{ m}$ (same reference level)?

     At $h=6\text{ m}$. Higher height means larger $h$ in $U_g = mgh$.

118. For the same spring, which stores more elastic potential energy: stretch $x$ or $2x$?

     Stretch $2x$ stores more ($4$ times as much). Energy depends on $x^2$, so doubling stretch quadruples energy.

119. Which object has greater gravitational potential energy if both have the same mass: higher or lower height?

     Higher height has greater $U_g$. Since $U_g = mgh$, greater height means greater energy.

120. What type of potential energy depends on the positions of charged particles relative to each other?

     Electric potential energy. Energy stored due to relative positions of charged objects.

121. Which is larger: the change in gravitational potential energy for a rise of $+2\text{ m}$ or $+5\text{ m}$ (same $m$)?

     The $+5\text{ m}$ rise. Larger height change means larger energy change.

122. Which spring stores more elastic potential energy: same $k$, displacements $x_1 > x_2$?

     The spring with displacement $x_1$. Energy depends on $x^2$, so larger displacement stores more.

123. Which has greater $U_s$: spring A $k=200\text{ N/m}, x=0.10\text{ m}$ or spring B $k=100\text{ N/m}, x=0.10\text{ m}$?

     Spring A. Spring A has twice the spring constant at same $x$.

124. Identify the correct comparison: same $k$; compression $x_1 > x_2$. Which has more $PE_e$?

     The spring with $x_1$ has greater $PE_e$. $PE_e \propto x^2$, so larger compression stores more energy.

125. If height doubles from $h$ to $2h$ (same $m$), how does $U_g$ change?

     $U_g$ doubles. Gravitational potential energy is directly proportional to height.

126. Which stores more elastic potential energy: same $k$, compressions $x=0.10\text{ m}$ or $x=0.20\text{ m}$?

     The $x=0.20\text{ m}$ compression. Since $U_s \propto x^2$, $(0.20)^2 = 4 \times (0.10)^2$.

127. Which object has greater gravitational potential energy if both are at the same height: larger or smaller mass?

     Larger mass has greater $U_g$. Since $U_g = mgh$, greater mass means greater energy.

128. What type of potential energy is stored in the positions of atoms within chemical substances (bonds)?

     Chemical potential energy. Energy stored in molecular bonds and atomic arrangements.

129. Which has greater $\Delta PE_g$: lifting $1\,\text{kg}$ by $4\,\text{m}$ or $2\,\text{kg}$ by $2\,\text{m}$ (same $g$)?

     They are equal ($m\Delta h = 4$ in both cases). Both give $\Delta PE_g = 4g$ joules since $mg\Delta h$ is the same.

130. What type of potential energy is stored when a spring is compressed or stretched?

     Elastic potential energy. Springs store energy when deformed from equilibrium.

131. What does the sign of $x$ (compression vs. stretch) do to $U_s = \frac{1}{2}kx^2$?

     $U_s$ is unchanged because $x^2$ is positive. Squaring $x$ eliminates the sign, so compression and stretch give same positive $U_s$.

132. Which spring stores more elastic potential energy: same $k$, displacements $x_1 > x_2$?

     The spring with displacement $x_1$. Larger displacement stores more elastic energy.

133. Choose the arrangement with greater gravitational potential energy: same object on a shelf or on the floor (same reference).

     On the shelf. Higher position means greater gravitational potential energy.

134. Two identical springs are stretched by $x$ and $3x$. Which has greater $U_s$?

     The spring stretched by $3x$ has greater $U_s$. Since $U_s \propto x^2$, tripling stretch gives 9× more energy.

135. Which arrangement has more $U_s$: same spring stretched $0.20\text{ m}$ or $0.30\text{ m}$?

     $0.30\text{ m}$ stretch. Larger stretch means larger $x^2$ in $U_s = \frac{1}{2}kx^2$.

136. Which stores more elastic potential energy: stretch $x$ or stretch $2x$ for the same spring constant $k$?

     Stretch $2x$ stores $4$ times as much. $U_s \propto x^2$, so doubling $x$ quadruples energy.

137. Choose the correct statement about reference level: is $U_g=0$ fixed at one height for all problems?

     No; the zero level for $U_g$ is chosen as a reference. Reference height can be chosen anywhere for convenience.

138. Identify the relationship between spring potential energy $U_s$ and stretch $x$ when $k$ is constant.

     $U_s$ is proportional to $x^2$. Doubling stretch quadruples spring potential energy.

139. What is the key comparison idea for spring potential energy when comparing different springs and stretches?

     Compare $k x^2$. Since $U_s = \frac{1}{2}kx^2$, compare the product of $k$ and $x^2$.

140. Which object has greater gravitational potential energy: same $m$, heights $h_1 > h_2$?

     The object at height $h_1$. Higher position means greater $U_g$ when mass is constant.

141. Which has greater total gravitational potential energy: two identical masses at height $h$ or one identical mass at height $2h$?

     They are equal ($2mgh = m g (2h)$). Total $U_g$ is same: $2 \times mgh = mg \times 2h$.

142. What is the formula for elastic potential energy stored in a spring?

     $PE_e = \frac{1}{2}kx^2$. Spring constant $k$ times displacement squared, divided by 2.

143. Which has more elastic potential energy: spring A with $k=200\text{ N/m}$ or spring B with $k=100\text{ N/m}$, same $x$?

     Spring A. Stiffer spring (higher $k$) stores more energy for same displacement.

144. Which arrangement has greater total gravitational potential energy: two identical masses each at height $h$ or one identical mass at height $h$?

     Two identical masses each at $h$ have greater total $U_g$. Total $U_g = 2(mgh)$ for two masses vs $mgh$ for one.

145. For the same stretch $x$, which spring stores more elastic potential energy: larger $k$ or smaller $k$?

     Larger $k$ stores more $U_s$. Stiffer springs (larger $k$) store more energy for same stretch.

146. Which system has more total potential energy: same spring, stretched $x$ and raised by $h$, or only raised by $h$?

     Stretched and raised: $PE_{tot}=mgh+\frac{1}{2}kx^2$. Total PE is the sum of gravitational and elastic contributions.

147. What is the reference level idea for gravitational potential energy $U_g$ in a system?

     $U_g$ is measured relative to a chosen zero height. Absolute $U_g$ depends on where you set zero height.

148. What is the formula for gravitational potential energy near Earth's surface?

     $PE_g = mgh$. Mass times gravity times height gives gravitational PE.

149. For the same stretch $x$, which stores more elastic potential energy: spring with $k=2k_0$ or $k=k_0$?

     Spring with $k=2k_0$ stores more $PE_e$. Stiffer spring (larger $k$) stores more energy for same stretch.

150. Which has greater total potential energy: $mgh$ only, or $mgh$ plus a stretched spring term $\frac{1}{2}kx^2$?

     $mgh + \frac{1}{2}kx^2$ is greater (for $x \neq 0$). Adding positive spring PE to gravitational PE increases total.

151. If height doubles from $h$ to $2h$ for the same mass, how does $U_g$ change?

     $U_g$ doubles. PE is proportional to height, so doubling $h$ doubles $U_g$.

152. What reference choice determines whether gravitational potential energy is positive, negative, or zero?

     The selected zero-height (reference level). Height is measured from a chosen reference point where $h = 0$.

153. Which has greater gravitational potential energy: mass $2m$ at height $h$ or mass $m$ at height $h$?

     Mass $2m$ at $h$ has greater $U_g$. Since $U_g = mgh$, doubling mass doubles the potential energy.

154. For the same spring, if stretch increases from $x$ to $3x$, by what factor does $U_s$ increase?

     $9$ times. Since $U_s \propto x^2$, tripling $x$ gives $3^2 = 9$ times more energy.

155. Which system arrangement has greater gravitational potential energy: two identical books stacked or the same books side-by-side on the floor?

     Stacked books. Stacked books have higher center of mass than side-by-side books.

156. State the formula for elastic potential energy of an ideal spring using $k$ and displacement $x$.

     $U_s = \frac{1}{2}kx^2$. Where $k$ is spring constant and $x$ is displacement from equilibrium.

157. What is the elastic potential energy formula for a spring stretched or compressed by distance $x$?

     $U_s = \frac{1}{2}kx^2$. Where $k$ is spring constant and $x$ is displacement.

158. Identify the relationship between $U_g$ and height $h$ when $m$ and $g$ are constant.

     $U_g$ is directly proportional to $h$. When mass and gravity are constant, $U_g$ increases linearly with height.

159. Which stores more elastic potential energy: stretch $x$ or stretch $2x$ (same spring)?

     Stretch $2x$ stores $4$ times as much $PE_e$. $PE_e \propto x^2$, so doubling $x$ quadruples energy.

160. Which system has more gravitational potential energy: same mass on a shelf or on the floor (shelf is higher)?

     The mass on the shelf has more $U_g$. Higher position gives more gravitational PE.

161. Which arrangement has greater $U_g$: masses $2m$ versus $m$ at the same height $h$?

     The object with mass $2m$. Doubling mass doubles $U_g$ since $U_g \propto m$.

162. For like charges, does electric potential energy increase or decrease as the charges move closer together?

     It increases as like charges move closer. Like charges repel, so forcing them closer stores energy.

163. Which object has greater gravitational potential energy: same mass at $2\text{ m}$ or at $5\text{ m}$?

     The object at $5\text{ m}$. Higher position means more gravitational potential energy.

164. If the reference level changes, what stays the same: absolute $U_g$ or the change $\Delta U_g$ between two heights?

     The change $\Delta U_g$ stays the same. PE differences between heights are independent of reference.

165. Which option has greater potential energy: same object at top of a hill or halfway down the hill (same reference)?

     Top of the hill has greater $PE_g$. Top position has maximum height, thus maximum $PE_g$.

166. Which spring stores more elastic potential energy: same $x$, spring constants $k_1 > k_2$?

     The spring with constant $k_1$. Stiffer springs store more energy for same displacement.

167. What is the meaning of "relative potential energy" when comparing two arrangements of the same system?

     It is the difference $\Delta U$ between arrangements. Compares energy states without depending on reference point.

168. What type of potential energy increases when an object is raised higher above the ground?

     Gravitational potential energy. Height increases the stored energy due to gravity's pull.

169. Which has greater elastic potential energy: same $k$, stretches $x_1 > x_2$?

     The spring stretched by $x_1$. Elastic potential energy increases with greater stretch or compression.

170. Identify which has greater gravitational potential energy: $1\,\text{kg}$ at $5\,\text{m}$ or $2\,\text{kg}$ at $2\,\text{m}$ (same $g$).

     $1\,\text{kg}$ at $5\,\text{m}$. $U_g = mgh$: $(1)(5) = 5$ vs $(2)(2) = 4$.

171. Which object has greater gravitational potential energy: $2\,\text{kg}$ at $3\,\text{m}$ or $3\,\text{kg}$ at $2\,\text{m}$ (same $g$)?

     They are equal ($m h$ is $6$ for both). $U_g = mgh$: $(2)(3) = 6$ and $(3)(2) = 6$.

172. Which arrangement has more $U_s$: $x=0.20\,\text{m}$ or $x=0.10\,\text{m}$ for the same spring?

     $x=0.20\,\text{m}$ has $4$ times the $U_s$. Quadrupling occurs because $(0.20)^2 = 4 \times (0.10)^2$.

173. Which option has greater elastic potential energy: spring stretched $x=0.10\text{ m}$ or $x=0.20\text{ m}$ (same $k$)?

     $x=0.20\text{ m}$ has greater $U_s$. Since $U_s \propto x^2$, larger stretch stores more energy.

174. A spring is compressed by $0.10\text{ m}$ and then by $0.20\text{ m}$. Which compression stores more $U_s$?

     The $0.20\text{ m}$ compression. Since $U_s \propto x^2$, doubling compression quadruples energy.

175. Identify the correct comparison: same $x$, spring constant increases from $k$ to $2k$; how does $U_s$ change?

     $U_s$ doubles. Since $k$ appears linearly in $U_s = \frac{1}{2}kx^2$, doubling $k$ doubles $U_s$.

176. Which has greater gravitational potential energy: same height $h$, masses $m_1 > m_2$?

     The object with mass $m_1$. Greater mass means greater $U_g$ when height is constant.

177. Which arrangement has more $U_g$: object A at $h=3\,\text{m}$ or object B at $h=1\,\text{m}$ (same $m$)?

     Object A at $3\,\text{m}$ has more $U_g$. Higher position means greater gravitational potential energy.

178. Which object has greater gravitational potential energy: same height $h$, masses $m_1 > m_2$?

     The object with mass $m_1$. Greater mass means greater gravitational potential energy.

179. Which system has greater gravitational potential energy: $m=2$ kg at $h=3$ m or $m=3$ kg at $h=2$ m (same $g$)?

     They are equal because $mh$ is the same. Both give $U_g = 6g$ J since $2 \times 3 = 3 \times 2 = 6$.

180. Which stores more elastic potential energy: spring with $k_1$ or $k_2$ stretched by the same $x$ if $k_2 > k_1$?

     The spring with $k_2$ stores more $U_s$. Stiffer spring (larger $k$) stores more energy for same stretch.

181. Which arrangement has more gravitational potential energy: same object on a shelf or on the floor?

     On the shelf. Higher position stores more gravitational potential energy.

182. Two objects are at the same height; one has mass $3m$ and one has mass $m$. Which has greater $U_g$?

     The object with mass $3m$ has greater $U_g$. Triple the mass triples the gravitational potential energy.

183. Which has greater gravitational potential energy: $m=2\,\text{kg}$ at $h=3\,\text{m}$ or $m=3\,\text{kg}$ at $h=2\,\text{m}$?

     They are equal (both have $m\cdot h = 6$). $U_g = mgh$: $2 \times 3 = 3 \times 2 = 6$ units.

184. Which has greater gravitational potential energy: same $m$, heights $h_1 > h_2$?

     The object at height $h_1$. Higher position means greater gravitational potential energy for same mass.

185. What happens to gravitational potential energy $U_g$ when height doubles from $h$ to $2h$ (same $m$)?

     $U_g$ doubles. Since $U_g = mgh$, doubling $h$ doubles $U_g$.

186. What is the relationship between work done on a spring and change in elastic potential energy?

     $W_{\text{on spring}} = \Delta U_s$. Work done stretching/compressing equals stored elastic PE.

187. If height doubles while mass stays constant, by what factor does $U_g$ change?

     $U_g$ doubles (factor of $2$). Since $U_g = mgh$, doubling $h$ doubles $U_g$.

188. State the formula for gravitational potential energy near Earth's surface.

     $U_g = mgh$. Where $m$ is mass, $g$ is gravity, and $h$ is height.

189. What is the formula for elastic potential energy stored in a spring with constant $k$ stretched by $x$?

     $U_s = \frac{1}{2}kx^2$. Elastic PE equals half the spring constant times stretch squared.

190. Identify the correct comparison: same $k$, stretch increases from $x$ to $2x$; how does $U_s$ change?

     $U_s$ becomes $4$ times larger. Since $x$ is squared, $(2x)^2 = 4x^2$, making $U_s$ four times larger.

191. What is elastic potential energy in a spring in terms of $k$ and stretch $x$?

     $U_s = \frac{1}{2}kx^2$. Energy stored in spring depends on stiffness $k$ and displacement squared.

192. Which has greater $U_s$: spring stretched by $x=+0.15\text{ m}$ or compressed by $x=-0.15\text{ m}$ (same $k$)?

     They are equal (depends on $x^2$). Only magnitude of displacement matters ($x^2$ is same).

193. Which has greater gravitational potential energy: $m=1\,\text{kg}$ at $h=5\,\text{m}$ or $m=2\,\text{kg}$ at $h=3\,\text{m}$?

     $m=2\,\text{kg}$ at $h=3\,\text{m}$ (larger $m\cdot h$). $U_g = mgh$: $2 \times 3 = 6 > 1 \times 5 = 5$ units.

194. If spring constant increases from $k$ to $3k$ (same $x$), how does $U_s$ change?

     $U_s$ triples. Elastic potential energy is directly proportional to spring constant.

195. Which has more elastic potential energy: same spring stretched $x$ or stretched $2x$?

     Stretched $2x$. Since $U_s \propto x^2$, doubling stretch quadruples energy.

196. If mass is tripled while height stays the same, how does $U_g$ change?

     $U_g$ triples. Since $U_g = mgh$, tripling $m$ triples the energy.

197. Identify the condition for zero elastic potential energy in a spring model.

     When $x = 0$ (unstretched length). Springs store no elastic energy at their natural, unstretched length.

198. What is gravitational potential energy in terms of $m$, $g$, and height $h$?

     $U_g = mgh$. Formula shows $U_g$ depends on mass, gravity, and height.

199. Which has greater spring potential energy: stretch $x$ or stretch $2x$ (same $k$)?

     Stretch $2x$; it has $4$ times the $U_s$. Since $U_s \propto x^2$, doubling $x$ gives $(2x)^2 = 4x^2$.

200. What is the unit of potential energy in the SI system?

     Joule (J). Energy is measured in joules, same as work.

201. What happens to $U_g$ if an object is moved downward by a height change of $-\Delta h$?

     $U_g$ decreases by $mg\Delta h$. Moving down reduces height, decreasing potential energy.

202. Which has greater gravitational potential energy: a book on a shelf or the same book on the floor (same reference)?

     The book on the shelf has greater $PE_g$. Higher position means larger $h$ in $PE_g = mgh$.

203. Which has greater $PE_g$: same mass on a $5\,\text{m}$ shelf or on a $2\,\text{m}$ shelf (same $g$)?

     On the $5\,\text{m}$ shelf. Greater height $h$ means greater $mgh$.

204. Which has more $U_g$: object A ($m$, $h$) or B ($2m$, $\frac{1}{2}h$) near Earth?

     They are equal. $U_g = mgh$: A has $mgh$, B has $2m \cdot \frac{h}{2} = mgh$.

205. What is the relationship between potential energy and the arrangement (position) of objects in a system?

     Potential energy depends on system arrangement and interactions. PE is stored energy based on object positions and forces.

206. Which has greater $U_s$: stretch $x$ with $k$ or stretch $\frac{1}{2}x$ with $4k$?

     They are equal. Both have $kx^2 = k cdot x^2 = 4k cdot (\frac{x}{2})^2 = kx^2$.

207. What is the elastic potential energy formula for an ideal spring?

     $U_s = \frac{1}{2}kx^2$. Spring potential energy equals half the spring constant times displacement squared.

208. Which has greater gravitational potential energy: same mass at height $2h$ or at height $h$?

     Height $2h$ has greater $PE_g$. PE is directly proportional to height in the formula $PE_g = mgh$.

209. Which stores more elastic potential energy: compression $x$ or tension $x$ for the same spring and same $|x|$?

     They are equal ($U_s$ depends on $x^2$). Only magnitude of displacement matters since $U_s \propto x^2$.

210. Which change increases gravitational potential energy the most: doubling $m$ or doubling $h$ (with $g$ constant)?

     Both increase $U_g$ by a factor of $2$. Since $U_g = mgh$, doubling either $m$ or $h$ doubles $U_g$.

211. Which stores more elastic potential energy: spring A with $k=200$ stretched $x$, or spring B with $k=100$ stretched $x$?

     Spring A. Spring A has twice the $k$ value, so twice the energy for same $x$.

212. Identify the correct comparison: same $m$, same $g$; height $h_1 > h_2$. Which has more $PE_g$?

     The object at $h_1$ has greater $PE_g$. Higher position means greater $mgh$ when $m$ and $g$ are constant.

213. Which has greater $U_g$: same object on the floor ($h=0$) or on a shelf ($h>0$) using the floor as reference?

     On the shelf ($h>0$). Shelf has positive height, floor has zero height.

214. If mass triples from $m$ to $3m$ (same $h$), how does $U_g$ change?

     $U_g$ triples. Gravitational potential energy is directly proportional to mass.

215. What is the key comparison idea for gravitational potential energy if $g$ is the same for all cases?

     Compare the product $m h$. Since $g$ is constant, only the product of mass and height matters.

216. Which comparison is correct for springs: if stretch changes from $x$ to $3x$, how does $U_s$ change?

     $U_s$ becomes $9$ times larger. Since $U_s \propto x^2$, tripling $x$ multiplies energy by $3^2$.

217. What is the formula for gravitational potential energy near Earth's surface?

     $U_g = mgh$. Mass times gravitational acceleration times height above reference.

218. Identify the energy type: energy stored due to position in Earth's gravitational field.

     Gravitational potential energy. Energy from object's position in gravitational field.

219. Which stores more energy in a spring: compressing it by $0.30\,\text{m}$ or by $0.10\,\text{m}$ (same $k$)?

     Compressing by $0.30\,\text{m}$ (9 times as much $PE_e$). $(0.30)^2 = 0.09$ vs $(0.10)^2 = 0.01$, so 9× more energy.

220. What is the gravitational potential energy formula near Earth for mass $m$ at height $h$?

     $PE_g = mgh$. Formula shows PE depends on mass, gravity, and height.

221. Which arrangement has greater gravitational potential energy: same $m$, heights $h_1$ and $h_2$ with $h_2>h_1$?

     The arrangement at $h_2$ has greater $U_g$. Greater height means greater gravitational potential energy.

222. What is the formula for elastic potential energy stored in a spring?

     $U_s = \frac{1}{2}kx^2$. Spring constant times displacement squared, divided by two.

223. What is the relationship between work done against gravity and change in gravitational potential energy?

     $W_{\text{against g}} = \Delta U_g$. Work done lifting an object equals its gain in gravitational PE.

224. Identify the reference-level idea: What does $h$ represent in $U_g=mgh$?

     Height measured relative to a chosen zero level. Height is always measured from a chosen reference point.

225. Identify the system arrangement with more stored elastic energy: rubber band stretched far or stretched slightly.

     Rubber band stretched far has more $PE_e$. More stretch means larger $x$ in $PE_e = \frac{1}{2}kx^2$.

226. Which change increases $PE_g$ the most for the same object: doubling $m$ or doubling $h$?

     Both increase $PE_g$ by a factor of $2$. Both $m$ and $h$ appear linearly in $PE_g = mgh$.

227. Which has greater gravitational potential energy: mass $m_1$ or $m_2$ at the same height $h$ if $m_2 > m_1$?

     The mass $m_2$ has greater $U_g$. Larger mass at same height has more PE since $U_g = mgh$.

228. Which has greater $PE_e$: same $k$ with compression magnitude $x$ or stretch magnitude $x$?

     They are equal (depends on $x^2$). Formula uses $x^2$, so direction doesn't matter, only magnitude.

229. Which has greater total potential energy: two identical masses each at height $h$ or one identical mass at height $h$?

     Two masses at height $h$ have greater total $PE_g$. Total PE is sum of individual PEs, so $2mgh > mgh$.

230. What happens to gravitational potential energy when an object falls downward (height decreases)?

     Gravitational potential energy decreases. Lower height means less PE since $U_g = mgh$.

Study these flashcards