Gravity Depends on Mass - Middle School Physical Science

1. Identify the correct comparison at constant $r$: if $m_1m_2$ is larger in Trial A than Trial B, which has larger $F_g$?

   Trial A has larger $F_g$. Larger mass product means larger gravitational force.

2. What happens to gravitational force if both masses are doubled and $r$ stays constant?

   $F$ becomes $4$ times larger. Doubling both masses quadruples their product $m_1 m_2$.

3. What happens to $F_g$ if $m_1$ is halved while $m_2$ and $r$ stay constant?

   $F_g$ is halved. Force is directly proportional to $m_1$, so halving it halves the force.

4. If $m_1$ triples and $m_2$ stays the same (constant $r$), what is the factor change in $F_g$?

   $F_g$ becomes $3$ times as large. Force is directly proportional to $m_1$, so tripling it triples $F_g$.

5. A line graph of $F$ vs. $m_1$ passes through the origin (constant $m_2,r$). What does this indicate?

   $F$ is directly proportional to $m_1$. Linear through origin confirms direct proportionality.

6. Identify the correct comparison: At the same $r$, which pair produces greater $F_g$: $(2,6)$ or $(3,3)$ (masses in kg)?

   $(2,6)$ produces greater $F_g$. Compare products: $2 \times 6 = 12 > 3 \times 3 = 9$.

7. What happens to gravitational force if $m_1$ is tripled while $m_2$ and $r$ stay constant?

   $F$ triples. Force is proportional to mass, so tripling mass triples force.

8. Two trials have equal $r$. Trial A has $(3m, 2m)$ and Trial B has $(4m, m)$. Which has larger $F_g$?

   Trial A. Trial A has mass product $6m^2$ vs Trial B's $4m^2$, so A has larger force.

9. Two data points: $(m_1,m_2,F)=(3,2,12)$ and $(3,6,?)$ with same $r$. What is $F$?

   $F=36$. $m_2$ triples from $2$ to $6$, so $F$ triples from $12$ to $36$.

10. If both masses double (same $r$), by what factor does $F_g$ change?

    $F_g$ increases by a factor of $4$. Doubling both masses quadruples their product $m_1m_2$.

11. A table shows $m_1$ doubles and $F_g$ quadruples (same $r$). Which mass change must also have occurred?

    $m_2$ also doubled. Quadrupling force with doubled $m_1$ means $m_2$ also doubled.

12. A data set shows $m_1m_2$ increases by a factor of $8$ at constant $r$. By what factor should $F_g$ increase?

    $F_g$ should increase by a factor of $8$. Force scales directly with mass product at constant distance.

13. If $m_1 = 2$ and $m_2 = 8$ at fixed $r$, what is $m_1 m_2$ (the mass factor for $F_g$)?

    $16$. Simply multiply the two mass values.

14. Identify the independent variable if a data set varies $m_2$ while $m_1$ and $r$ are held constant.

    Independent variable: $m_2$. The variable being changed in the experiment is the independent variable.

15. What happens to gravitational force if $m_1$ is doubled while $m_2$ and $r$ stay constant?

    $F$ doubles. Since $F \propto m_1$, doubling $m_1$ doubles the force.

16. Two trials have the same $r$. Trial A has $(m_1,m_2)=(2,3)$ and Trial B has $(m_1,m_2)=(1,12)$. Which has larger $F_g$?

    Trial B (product $m_1m_2$ is larger). Compare products: $2 \times 3 = 6$ vs $1 \times 12 = 12$.

17. Identify the relationship between $F_g$ and the masses when distance is constant: direct or inverse?

    Directly proportional to $m_1$ and to $m_2$. Force increases when either mass increases (positive relationship).

18. If both masses double while $r$ stays constant, what happens to $F_g$?

    $F_g$ becomes $4$ times as large. Since $F_g \propto m_1 m_2$, doubling both gives $2 \times 2 = 4$ times the force.

19. If $m_1$ becomes $\frac{1}{2}m_1$ and $m_2$ stays the same (same $r$), how does $F_g$ change?

    $F_g$ becomes $\frac{1}{2}$ of the original. Halving one mass halves the force proportionally.

20. Two trials have equal $r$. Trial A: $m_1m_2=6$; Trial B: $m_1m_2=18$. What is $\frac{F_B}{F_A}$?

    $3$. Force ratio equals mass product ratio: $\frac{18}{6} = 3$.

21. A graph plots $F_g$ versus $m_1$ with $m_2$ and $r$ constant. What is the expected graph shape?

    A straight line through the origin. Direct proportionality creates a linear relationship passing through origin.

22. Two interactions have the same $r$. Interaction A uses $(m_1,m_2)=(4,5)$ and B uses $(2,10)$. Which has larger $F_g$?

    They are equal. Both have same mass product: $4 \times 5 = 2 \times 10 = 20$.

23. Which statement correctly describes how gravitational force depends on mass when distance is held constant?

    $F_g$ increases in direct proportion to $m_1m_2$. Summarizes the key relationship from Newton's law.

24. If $F_g$ is $20\ \text{N}$ when $m_1 = 5\ \text{kg}$, what is $F_g$ when $m_1 = 10\ \text{kg}$ (same $m_2$, $r$)?

    $40\ \text{N}$. Doubling $m_1$ doubles the force when other variables are constant.

25. If $F_g$ is $18\ \text{N}$ for masses $(3\ \text{kg}, 6\ \text{kg})$, what is $F_g$ for $(3\ \text{kg}, 2\ \text{kg})$ at same $r$?

    $6\ \text{N}$. $m_2$ changes from $6$ to $2$ kg, reducing force by factor $\frac{1}{3}$: $18 \times \frac{1}{3} = 6$ N.

26. If $m_1$ increases by a factor of $5$ and $m_2$ decreases by a factor of $5$ with constant $r$, what happens to $F_g$?

    $F_g$ stays the same. Mass product remains unchanged: $5m_1 \times \frac{m_2}{5} = m_1m_2$.

27. If $m_1 m_2$ increases from $6$ to $18$ with constant $r$, by what factor does $F_g$ change?

    $3$ times larger. Force scales with mass product: $\frac{18}{6} = 3$.

28. If $m_1$ increases by a factor of $5$ (with $m_2$ and $r$ constant), how does $F_g$ change?

    $F_g$ increases by a factor of $5$. Direct proportionality to $m_1$ means force scales by the same factor.

29. If $m_1$ triples while $m_2$ and $r$ stay constant, what is the factor change in $F_g$?

    $F_g$ becomes $3$ times as large. Direct proportionality means tripling one mass triples the force.

30. Choose the correct statement about forces: object $1$ pulls on object $2$ with $F_g$, and object $2$ pulls on object $1$ with what?

    The same magnitude $F_g$ (opposite direction). Newton's third law: forces between objects are equal and opposite.

31. If both masses double, what is the factor change in $F_g$ (with $r$ constant)?

    $F_g$ becomes $4$ times as large. Force is proportional to product $m_1m_2$, so $2 \times 2 = 4$.

32. Which conclusion matches the data: $m_1 m_2$ doubles and $F$ doubles (with $r$ constant)?

    $F \propto m_1 m_2$. Equal scaling confirms direct proportionality.

33. If a graph plots $F_g$ (y-axis) versus $m_1m_2$ (x-axis) with constant $r$, what shape should the data follow?

    A straight line through the origin. Direct proportionality creates a linear relationship passing through $(0,0)$.

34. A data table shows $r$ constant. If $m_1$ doubles and measured $F_g$ does not change, what is the correct conclusion?

    The data are inconsistent with gravity-only dependence. If only gravity acts, doubling mass must double force; unchanged force indicates other factors.

35. What is the ratio $\frac{F_2}{F_1}$ if $m_1$ changes from $m_1$ to $3m_1$ (same $m_2,r$)?

    $\frac{F_2}{F_1}=3$. Tripling one mass triples the force proportionally.

36. If $m_1$ doubles while $m_2$ and $r$ stay constant, how does $F_g$ change?

    $F_g$ doubles. Since $F_g \propto m_1$, doubling $m_1$ doubles the force.

37. If $m_1$ triples while $m_2$ and $r$ stay constant, by what factor does $F_g$ change?

    $F_g$ becomes $3$ times as large. Tripling one mass triples the product $m_1m_2$ and thus $F_g$.

38. A graph plots $F_g$ versus $m_1m_2$ with constant $r$. What is the expected relationship?

    Linear: $F_g$ increases directly with $m_1m_2$. Direct proportionality means force increases linearly with mass product.

39. If $F_g$ is $24\,\text{N}$ for masses $(m_1,m_2)$ at fixed $r$, what is $F_g$ when $m_1$ is quadrupled?

    $96\,\text{N}$. Quadrupling $m_1$ quadruples force: $24 \times 4 = 96$.

40. Which graph best represents $F_g$ versus $m_1$ when $m_2$ and $r$ are constant?

    A straight line through the origin. Direct proportionality creates a linear relationship passing through zero.

41. Choose the correct statement for constant $r$: $F_g$ is directly proportional to which quantity?

    The product $m_1m_2$. At constant distance, gravitational force varies with mass product.

42. Identify the missing value: $m_1=2,m_2=4,F=8$ and then $m_1=5,m_2=4,F=?$ (same $r$).

    $F=20$. $m_1$ increases by $2.5$ ($5/2$), so $F$ increases by $2.5$ to $20$.

43. Choose the statement that must be true if $r$ is constant: $F_g$ depends on the sum $m_1+m_2$ or the product $m_1m_2$?

    The product $m_1m_2$. Formula shows $F_g \propto m_1m_2$, not $m_1 + m_2$.

44. Identify the mass quantity you should compute from data to test the mass dependence of gravity at fixed $r$.

    Compute the product $m_1m_2$. The product reveals the combined mass effect on gravitational force.

45. If $F_g$ is $12\,\text{N}$ when $m_1 m_2 = 24$ at fixed $r$, what is $F_g$ when $m_1 m_2 = 48$?

    $24\,\text{N}$. Force doubles when mass product doubles: $\frac{48}{24} = 2$, so $12 \times 2 = 24\,\text{N}$.

46. A graph shows $F_g$ vs. $m_1$ is a straight line through the origin (constant $m_2$ and $r$). What does this mean?

    $F_g$ is directly proportional to $m_1$. A straight line through origin indicates direct proportionality.

47. What happens to $F_g$ if $m_1$ is tripled while $m_2$ and $r$ stay the same?

    $F_g$ triples. Force is directly proportional to $m_1$, so tripling it triples the force.

48. If the product $m_1m_2$ increases from $12$ to $48$ while $r$ stays constant, by what factor does $F_g$ change?

    $F_g$ becomes $4$ times as large. The ratio $\frac{48}{12} = 4$ shows force increases by factor of $4$.

49. A data table shows $F_g$ stays constant when $r$ is constant but $m_1$ increases. What must happen to $m_2$?

    $m_2$ must decrease in inverse proportion to $m_1$. To keep $m_1m_2$ constant, $m_2$ must decrease as $m_1$ increases.

50. If $m_1$ increases by a factor of $5$ (constant $m_2$ and $r$), what is the factor change in $F_g$?

    $F_g$ increases by a factor of $5$. Direct proportionality means force increases by the same factor as mass.

51. Two trials have equal $r$. Trial A has masses $(m, m)$ and Trial B has $(2m, m)$. Which trial has larger $F_g$?

    Trial B. Trial B has mass product $2m^2$ vs Trial A's $m^2$, so B has larger force.

52. What happens to $F_g$ if both $m_1$ and $m_2$ are doubled while $r$ stays the same?

    $F_g$ becomes $4$ times larger. Force depends on the product $m_1 m_2$, so doubling both gives $2 \times 2 = 4$ times the force.

53. If $m_1$ doubles while $m_2$ and $r$ stay constant, what happens to $F_g$?

    $F_g$ doubles. Since $F_g \propto m_1$, doubling $m_1$ doubles the force.

54. If measured $F_g$ does not scale with $m_1m_2$ as expected, what variable most likely was not held constant?

    The distance $r$. Distance appears in denominator, so varying $r$ disrupts the mass relationship.

55. Two trials keep $r$ the same. Trial A has $m_1m_2=12$, Trial B has $m_1m_2=20$. Which trial has larger $F_g$?

    Trial B. Larger mass product ($20 > 12$) means larger gravitational force.

56. If $m_1$ doubles and $m_2$ becomes $\frac{1}{2}m_2$ (same $r$), how does $F_g$ change?

    $F_g$ stays the same. Product $m_1m_2$ remains unchanged: $2m_1 \times \frac{1}{2}m_2 = m_1m_2$.

57. If both masses $m_1$ and $m_2$ double while $r$ stays constant, what happens to $F_g$?

    $F_g$ becomes $4$ times as large. Doubling both masses quadruples their product $(2 \times 2 = 4)$.

58. If $m_1$ doubles while $m_2$ and $r$ stay constant, what happens to $F_g$?

    $F_g$ doubles. Since $F_g \propto m_1m_2$, doubling one mass doubles the force.

59. Which proportionality statement correctly describes how $F_g$ depends on mass?

    $F_g \propto m_1 m_2$. The formula shows force is proportional to the product of both masses.

60. If $m_2$ is cut to $\frac{1}{2}$ while $m_1$ and $r$ stay constant, what happens to $F_g$?

    $F_g$ becomes $\frac{1}{2}$ as large. Halving one mass halves the product $m_1m_2$ and thus $F_g$.

61. A data table shows $m_1$ doubles and $m_2$ triples at constant $r$; what should $F_g$ do?

    $F_g$ becomes $6$ times as large. The mass product increases by factor $2 \times 3 = 6$.

62. What happens to $F_g$ if both masses double while distance $r$ stays constant?

    $F_g$ becomes $4$ times as large. Doubling both masses quadruples their product, quadrupling the force.

63. If $F_g$ is $30\,\text{N}$ when $m_1 m_2 = 15$ at fixed $r$, what is $F_g$ when $m_1 m_2 = 5$?

    $10\,\text{N}$. Force scales with mass product: $\frac{5}{15} = \frac{1}{3}$, so $30 \times \frac{1}{3} = 10\,\text{N}$.

64. What is the gravitational force formula showing how force depends on two masses?

    $F_g = G\frac{m_1 m_2}{r^2}$. Newton's law shows force is proportional to both masses and inversely to distance squared.

65. If $F_g$ is $12\ \text{N}$ for masses $m_1$ and $m_2$ at fixed $r$, what is $F_g$ when $m_2$ becomes $3m_2$?

    $36\ \text{N}$. Tripling one mass triples the force: $3 \times 12 = 36$ N.

66. A data table shows $F_g$ doubles when only $m_2$ doubles (constant $r$). What mass relationship does this support?

    $F_g$ is directly proportional to $m_2$. Doubling $m_2$ doubles $F_g$ confirms direct proportionality.

67. Identify the correct conclusion: If $m_1$ is $10$ times $m_2$ at the same $r$, which force is larger, on $m_1$ or on $m_2$?

    Neither; the forces are equal in magnitude. Newton's third law: gravitational forces are equal and opposite pairs.

68. If $m_1 m_2$ is constant while $r$ is constant, what can you conclude about $F_g$?

    $F_g$ is constant. Since $F_g \propto m_1 m_2$ at fixed $r$, constant product means constant force.

69. What happens to $F_g$ if $m_1$ is halved while $m_2$ and $r$ stay the same?

    $F_g$ is cut in half. Force is directly proportional to $m_1$, so halving it halves the force.

70. If $m_1=2\ \text{kg}$ and $m_2=5\ \text{kg}$ at fixed $r$, what is the proportional mass factor $m_1m_2$?

    $10\ \text{kg}^2$. Multiply the masses: $2 \times 5 = 10$ with units $\text{kg}^2$.

71. Two trials keep $r$ constant. Trial A has $(m_1,m_2)$ and Trial B has $(2m_1,m_2)$. Which trial has larger $F_g$?

    Trial B. Trial B has twice the mass product, so twice the force.

72. Two data points: $(m_1,m_2,F)=(2,5,10)$ and $(4,5,?)$ with same $r$. What is $F$?

    $F=20$. $m_1$ doubles from $2$ to $4$, so $F$ doubles from $10$ to $20$.

73. What is the proportionality statement for mass dependence when $r$ is constant?

    $F_g \propto m_1m_2$. Mathematical notation showing force scales with mass product.

74. Two trials have the same $r$. Trial A has $m_1m_2=18$ and Trial B has $m_1m_2=6$. What is $\frac{F_A}{F_B}$?

    $\frac{F_A}{F_B}=3$. Force ratio equals mass product ratio: $\frac{18}{6} = 3$.

75. If $F_g$ is $50\,\text{N}$ at fixed $r$, what is the new $F_g$ when $m_1$ is doubled and $m_2$ is halved?

    $50\,\text{N}$. Doubling one mass and halving the other leaves force unchanged.

76. Identify the missing value: $m_1=6,m_2=3,F=18$ and then $m_1=6,m_2=1,F=?$ (same $r$).

    $F=6$. $m_2$ decreases by factor of $3$, so $F$ decreases to $6$.

77. Identify the correct ratio at constant $r$: $\frac{F_2}{F_1} = \frac{m_{1,2}m_{2,2}}{m_{1,1}m_{2,1}}$ or $\frac{m_{1,2}+m_{2,2}}{m_{1,1}+m_{2,1}}$?

    $\frac{F_2}{F_1}=\frac{m_{1,2}m_{2,2}}{m_{1,1}m_{2,1}}$. Force ratio equals mass product ratio when distance is constant.

78. If $m_1$ becomes $4$ times larger and $m_2$ becomes $\frac{1}{2}$ as large, with $r$ constant, what happens to $F_g$?

    $F_g$ becomes $2$ times as large. The product changes by factor $4 \times \frac{1}{2} = 2$.

79. If $m_1$ becomes $10$ times larger while $m_2$ and $r$ stay constant, what happens to $F_g$?

    $F_g$ becomes $10$ times as large. The force scales directly with mass, so $10\times$ mass gives $10\times$ force.

80. Two trials have the same $r$. Trial A uses masses $(2,6)$ and Trial B uses $(3,8)$. What is $\frac{F_B}{F_A}$?

    $\frac{F_B}{F_A} = 2$. $\frac{3 \times 8}{2 \times 6} = \frac{24}{12} = 2$

81. A graph shows $F_g$ values $4, 8, 12\,\text{N}$ for $m_1$ values $1, 2, 3$ (constant $m_2$ and $r$). What is $F_g$ at $m_1=5$?

    $20\,\text{N}$. Linear relationship gives $4\,\text{N}$ per unit $m_1$, so $5 \times 4 = 20\,\text{N}$.

82. What is the gravitational force ratio $\frac{F_2}{F_1}$ if $m_1$ changes from $2$ to $6$ (others constant)?

    $\frac{F_2}{F_1}=3$. $m_1$ increases by factor of $3$, so force triples.

83. If $m_1$ increases by a factor of $5$ and $m_2$ decreases by a factor of $2$ (same $r$), what is the net factor on $F_g$?

    $\frac{5}{2}$ times larger. Net factor is $\frac{5}{2}$ since $F_g$ depends on product of masses.

84. If two trials have the same $r$ and the same $m_1m_2$, what should be true about $F_g$?

    The forces should be equal, $F_{g^1}=F_{g^2}$. Same distance and mass product yield identical gravitational forces.

85. Which graph best shows $F_g$ vs. $m_1$ when $m_2$ and $r$ are constant: a straight line through the origin or a curve?

    A straight line through the origin. Direct proportionality creates a linear relationship passing through zero.

86. What happens to $F_g$ if $m_1$ is doubled while $m_2$ and $r$ stay the same?

    $F_g$ doubles. Force is directly proportional to each mass, so doubling one mass doubles the force.

87. With constant $r$, if $m_1m_2$ is multiplied by $\frac{1}{4}$, what happens to $F_g$?

    $F_g$ is multiplied by $\frac{1}{4}$. Force scales with mass product, so it's also multiplied by $\frac{1}{4}$.

88. If both $m_1$ and $m_2$ double while $r$ stays constant, what happens to $F_g$?

    $F_g$ becomes $4$ times as large. Doubling both masses quadruples their product $(2 \times 2 = 4)$.

89. What does it mean if data show $F_g$ doubles when $m_1$ doubles (same $m_2,r$)?

    $F_g \propto m_1$ for constant $m_2$ and $r$. Direct proportionality means force scales linearly with mass.

90. If the product $m_1m_2$ increases from $10$ to $30$ and $r$ is constant, what happens to $F_g$?

    $F_g$ becomes $3$ times as large. Force is proportional to mass product: $30/10 = 3$ times larger.

91. If a graph of $F_g$ versus $m_1m_2$ is linear through the origin (constant $r$), what is the conclusion?

    $F_g \propto m_1m_2$ when $r$ is constant. Linear graph through origin confirms direct proportionality.

92. If $m_1$ doubles and $m_2$ halves while $r$ stays constant, what happens to $F_g$?

    $F_g$ stays the same. The product $m_1 m_2$ remains unchanged: $2m_1 \times \frac{m_2}{2} = m_1 m_2$.

93. Identify the relationship shown by a straight-line graph of $F_g$ versus $m_1$ when $m_2$ and $r$ are constant.

    Directly proportional (linear through the origin). Straight line through origin indicates direct proportionality.

94. Two experiments keep $r$ constant. Exp 1: $m_1=2$, $m_2=6$. Exp 2: $m_1=3$, $m_2=4$. Which has larger $F_g$?

    They are equal. Both have same mass product: $2 \times 6 = 3 \times 4 = 12$.

95. Which relationship between force and mass product is correct when $r$ is constant: $F_g\propto m_1+m_2$ or $F_g\propto m_1m_2$?

    $F_g\propto m_1m_2$. Force depends on the product of masses, not their sum.

96. Which mass change increases $F_g$ more: doubling $m_1$ or doubling both $m_1$ and $m_2$ (same $r$)?

    Doubling both $m_1$ and $m_2$. Doubling both masses ($4\times$ increase) exceeds doubling one ($2\times$).

97. If $m_1m_2$ increases from $12$ to $36$ at constant $r$, by what factor does $F_g$ change?

    $F_g$ increases by a factor of $3$. $\frac{36}{12} = 3$, so force triples when mass product triples.

98. If the product $m_1m_2$ increases from $12$ to $36$ with $r$ constant, what happens to $F_g$?

    $F_g$ becomes $3$ times as large. Force ratio equals mass product ratio: $\frac{36}{12} = 3$.

99. A table shows $m_1 m_2$ increases from $12$ to $36$ with constant $r$. What is $\frac{F_2}{F_1}$?

    $\frac{F_2}{F_1} = 3$. Force ratio equals mass product ratio: $\frac{36}{12} = 3$.

100. Which quantity must you keep constant to isolate how $F_g$ changes with mass in data?

     Distance between centers, $r$. Keeping $r$ constant isolates the effect of mass changes on $F_g$.

101. If the product $m_1 m_2$ increases from $12$ to $48$ while $r$ is constant, what happens to $F$?

     $F$ becomes $4$ times larger. Product increases by factor of $4$ ($48/12$), so force quadruples.

102. What is the ratio $\frac{F_2}{F_1}$ if masses change to $(\frac{1}{4}m_1,4m_2)$ with the same $r$?

     $\frac{F_2}{F_1}=1$. Product remains same: $\frac{1}{4}m_1 \times 4m_2 = m_1m_2$.

103. Two trials keep $r$ constant. Trial A has masses $(2,6)$ and Trial B has masses $(3,4)$. Which trial has larger $F_g$?

     Equal $F_g$ (both have $m_1 m_2 = 12$). Force depends on mass product, which equals $12$ for both trials.

104. What happens to $F_g$ if $m_1$ doubles while $m_2$ and $r$ stay constant?

     $F_g$ doubles. Force is directly proportional to each mass, so doubling one doubles the force.

105. If $F_g$ is $40\ \text{N}$ at fixed $r$, what is the new $F_g$ when $m_1$ becomes $\frac{1}{2}m_1$ and $m_2$ becomes $2m_2$?

     $40\ \text{N}$. Mass product unchanged: $\frac{1}{2} \times 2 = 1$, so force stays same.

106. A graph plots $F_g$ vs. $m_1$ with $m_2$ and $r$ constant. What should the graph look like?

     A straight line through the origin. Direct proportion always graphs as a line through origin.

107. What happens to $F_g$ if $m_1$ is tripled while $m_2$ and $r$ stay constant?

     $F_g$ triples. Force is directly proportional to $m_1$, so tripling it triples the force.

108. A data set shows $F_g$ does not change when $m_1$ changes. Which variable was most likely not actually held constant?

     $r$ was not held constant. If force doesn't change when mass changes, distance must have varied.

109. Which statement correctly compares forces: $F_g$ between $(m_1,m_2)$ and between $(m_2,m_1)$ at the same $r$?

     They are equal. Newton's third law: gravitational forces between objects are equal and opposite.

110. If $F_g$ is $12\ \text{N}$ when $m_1m_2=8$ at fixed $r$, what is $F_g$ when $m_1m_2=2$?

     $3\ \text{N}$. Force scales with mass product: $12 \times \frac{2}{8} = 3$.

111. Which trial has the greater gravitational force if $r$ is the same: Trial A has $m_1m_2=20$, Trial B has $m_1m_2=15$?

     Trial A. Larger mass product ($20 > 15$) produces greater gravitational force.

112. Identify the dependent variable in a table studying how changing mass affects gravitational force.

     Dependent variable: $F_g$. The measured outcome (force) is the dependent variable.

113. If $m_1 m_2$ decreases from $20$ to $5$ with constant $r$, by what factor does $F_g$ change?

     $\frac{1}{4}$ as large. Force scales with mass product: $\frac{5}{20} = \frac{1}{4}$.

114. Choose the correct statement: Is $F_g$ proportional to $m_1+m_2$ or to $m_1m_2$ (with $r$ constant)?

     $F_g$ is proportional to $m_1m_2$. The formula shows force depends on product, not sum, of masses.

115. Two trials have the same $r$. If Trial A has $m_1m_2 = 15$ and Trial B has $m_1m_2 = 45$, what is $\frac{F_B}{F_A}$?

     $\frac{F_B}{F_A} = 3$. $\frac{45}{15} = 3$, so force ratio equals mass product ratio.

116. What type of relationship between gravitational force and mass is shown by $F_g = G\frac{m_1m_2}{r^2}$ when $r$ is constant?

     Directly proportional to the product $m_1m_2$. When $r$ is fixed, $F_g$ increases linearly with the product of masses.

117. A data table shows $m_1$ doubles while $m_2$ and $r$ stay constant. What pattern should the $F_g$ column show?

     $F_g$ doubles each time $m_1$ doubles. Linear relationship means equal increments produce equal changes.

118. Identify the best conclusion from data: If $m_1 m_2$ quadruples at constant $r$, what should happen to $F_g$?

     $F_g$ should quadruple. Force is directly proportional to mass product at constant distance.

119. What is the ratio $\frac{F_2}{F_1}$ if only $m_2$ changes from $m_2$ to $2m_2$ (same $m_1,r$)?

     $\frac{F_2}{F_1}=2$. Force doubles when one mass doubles, keeping others constant.

120. If $m_1$ doubles and $m_2$ and $r$ stay constant, what happens to $F_g$?

     $F_g$ doubles. Since $F_g \propto m_1$, doubling $m_1$ doubles the force.

121. State the formula for gravitational force between two objects (Newton's law of gravitation).

     $F_g = G\frac{m_1m_2}{r^2}$. Newton's universal law of gravitation relates force to masses and distance.

122. If $m_1$ triples and $m_2$ stays the same (same $r$), by what factor does $F_g$ change?

     $F_g$ increases by a factor of $3$. Force is proportional to $m_1$, so tripling it triples $F_g$.

123. A table shows $m_1$ doubles while $m_2,r$ stay constant. Which pattern should $F$ follow?

     $F$ should double each time. Direct proportionality means equal scaling.

124. Two trials keep $r$ constant. Trial A has $(m_1,m_2)$ and Trial B has $(m_1,3m_2)$. What is $\frac{F_B}{F_A}$?

     $\frac{F_B}{F_A}=3$. Force ratio equals mass product ratio: $\frac{m_1 \cdot 3m_2}{m_1 \cdot m_2} = 3$.

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