Show Me The Physics
AP Gravity

Universal Gravitation Module

I. Defining Gravity

A. Defined: of between any two objects in the universe.

$G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$

$m_1, m_2 = \text{masses attracting each other}$ | $r = \text{distance between centers}$

B. Relationships

1. Relationship between Force ($F$) and distance ($r$): . Plot shape: .

2. Tripling the distance causes the Force to change by: .

3. Relationship between Force ($F$) and mass ($m$): . Plot shape: .

Ex 1) Two objects a distance $r$ apart have force $F$. Masses changed to $2m_1$ and $2m_2$, and distance changed to $4r$. New force?

II. Satellite & Weight

Ex 2) If the gravitational force on satellite A is 160,000 N, what is the force on satellite B (same mass, double distance from center)?

Ex 3) A rocket weighs 18,000 N at surface. If it rises to a height where its distance from the center is 3 Earth radii, its weight is...

III. Surface Gravity

Earth Gravity Data: NY (9.803 m/s²) / Denver (9.796 m/s²) / North Pole (9.832 m/s²)

What explains these differences?

Formula for surface gravity: $g = \frac{G M_p}{r^2}$.

Ex 4) Mars mass is 1/10 Earth, diameter is 1/2 Earth. $g_{Mars} \approx $ .

Practice Problems

28. A 2100-kg spacecraft is in orbit 2.0 Earth radii from the Earth's center. Calculate the force of gravity on the spacecraft.

32. A hypothetical planet has a radius 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity on its surface?

IV. Orbital Velocity Proofs

Ex 5) Prove orbital velocity is independent of satellite mass using the derivation above.

a) Orbital $v$ and Period $T$ are determined only by: .

b) The smaller the radius, the the orbital velocity and the period $T$.

V. Complete MCQ Bank (1 - 37)

1. 1. The gravitational force between two objects is proportional to:
   A) the distance between them

   B) the square of the distance between them

   C) the product of the masses of the objects

   D) the square of the product of the masses
2. 2. Two objects attract each other. If the distance between their centers is cut in half, the force between them:
   A) is cut to 1/4

   B) is cut in half

   C) doubles

   D) quadruples
3. 3. Two objects a distance $r$ apart have a force $F$ between them. If the masses are changed to $2m_1$ and $2m_2$ and the distance is changed to $4r$, the new force is:
   A) F/16

   B) F/4

   C) 16F

   D) 4F
4. 4. As a rocket moves away from the Earth, its weight:
   A) increases

   B) decreases

   C) remains the same

   D) depends on its speed
5. 5. A planet has half the mass of Earth and twice the radius. The acceleration due to gravity on its surface is:
   A) g

   B) g/2

   C) g/4

   D) g/8
6. 6. Planet B has twice the mass of planet A, and both have the same surface gravity. If planet A has radius $r$, planet B has radius:
   A) 0.707r

   B) r

   C) 1.41r

   D) 2r
7. 7. If the mass of the Earth tripled and the radius doubled, a 10. kg object would have a weight of:
   A) 1.5mg

   B) 3/4mg

   C) 3/2mg

   D) 6mg
8. 8. A star has 4 times the mass of the Sun. A planet at 1 AU around it will have a period:
   A) 1/4 as much

   B) 1/2 as much

   C) twice as much

   D) four times as much
9. 9. The gravitational pull of the Sun on the Earth is:
   A) same as Earth on Sun

   B) more than Earth on Sun

   C) less than Earth on Sun

   D) zero
10. 10. The value of the universal gravitational constant G was first measured by:
    A) Newton

    B) Cavendish

    C) Galileo

    D) Einstein
11. 11. If the distance between two objects is halved, the force of gravity becomes:
    A) 4F

    B) 2F

    C) F/2

    D) F/4
12. 12. A planet has 4x the mass and 2x the radius of Earth. A 12 N jar on Earth weighs:
    A) 6 N

    B) 12 N

    C) 24 N

    D) 48 N
13. 13. At a distance of 3R above the Earth's surface, the acceleration of gravity is:
    A) zero

    B) same as surface

    C) g/9

    D) g/16
14. 14. What is the speed for a low circular orbit skimming the Earth's surface? ($R=6.4 \times 10^6$m)
    A) 5.9 km/s

    B) 6.9 km/s

    C) 7.9 km/s

    D) 8.9 km/s
15. 15. Moon A is at radius r, Moon B is at 4r. A takes 20 days. Moon B takes:
    A) 20 d

    B) 80 d

    C) 160 d

    D) 320 d
16. 16. Two objects with masses $2m$ and $2m$ are distance $2r$ apart. New Force is:
    A) F

    B) 2F

    C) 4F

    D) F/2
17. 17. An astronaut on the Space Station weighs:
    A) zero

    B) approx 95% of Earth weight

    C) approx 5% of Earth weight

    D) same as Earth weight
18. 18. Surface gravity on a planet with mass 2M and radius 2R compared to Earth:
    A) g

    B) 2g

    C) g/2

    D) g/4
19. 19. Astronaut at $1 R_e$ height above surface. Gravity compared to surface is:
    A) g

    B) g/2

    C) g/4

    D) zero
20. 20. Planet with 1/4 mass and 1/2 radius of Earth. Weight compared to Earth:
    A) same

    B) twice

    C) half

    D) 1/4
21. 21. Weighs 100 N on Earth. Weighs 100 N on a planet with 4x mass and radius:
    A) r

    B) 2r

    C) 4r

    D) 16r
22. 22. An object weighs 432 N on Earth. At a height of 3R above surface, weight is:
    A) 432 N

    B) 48 N

    C) 27 N

    D) 0 N
23. 23. If the distance between two masses is quadrupled, the force is:
    A) 1/4

    B) 1/16

    C) 4x

    D) 16x
24. 24. Symmetric planet has 2x mass and 2x radius of Earth. 12 N object weighs:
    A) 6 N

    B) 12 N

    C) 24 N

    D) 48 N
25. 25. Mass of moon is 1/81 of Earth. The pull of Earth on moon is:
    A) 81x Moon on Earth

    B) same as Moon on Earth

    C) 1/81 Moon on Earth

    D) zero
26. 26. Pull of a satellite on Earth is:
    A) zero

    B) less than Earth on satellite

    C) same as Earth on satellite

    D) depends on orbit height
27. 27. Gravitational force on a satellite in circular orbit is directed:
    A) tangent to orbit

    B) opposite motion

    C) away from Earth

    D) toward the center
28. 28. Which satellite has the lowest orbital speed?
    A) mass m/2, radius R/2

    B) mass m, radius R/2

    C) mass m, radius R

    D) mass m, radius 2R
29. 29. If the satellite mass is doubled in the same orbit, the speed is:
    A) the same

    B) doubled

    C) halved

    D) quadrupled
30. 30. Planet mass x10 and radius x5. Surface gravity compared to Earth is:
    A) 2g

    B) 0.4g

    C) 5g

    D) 0.1g
31. 31. Gravitational field lines point toward:
    A) center of mass

    B) infinity

    C) tangent direction

    D) random direction
32. 32. Astronaut in space shuttle feels weightless because:
    A) there is no gravity

    B) center of gravity is zero

    C) in free fall

    D) shielded from gravity
33. 33. Speed of a satellite depends on:
    A) mass of satellite

    B) mass of central body

    C) both

    D) neither
34. 34. Weightlessness in the space station is due to:
    A) lack of gravity

    B) vacuum environment

    C) free fall motion

    D) zero air pressure
35. 35. Object weighs 10N on Earth. On planet with same radius but 2x mass, weight:
    A) 5 N

    B) 10 N

    C) 20 N

    D) 40 N
36. 36. Kepler's 3rd law relates orbital period to:
    A) orbital mass

    B) orbital radius

    C) eccentricity

    D) orbital area
37. 37. Moon gravity 1/6 Earth, radius 1/4 Earth. Moon mass is:
    A) 1/6

    B) 1/16

    C) 1/24

    D) 1/96

VI. Extended Analysis

1. Lunar Eclipse Net Force

During a lunar eclipse, the Moon ($m=7.4 \times 10^{22}$ kg, $r=3.8 \times 10^8$ m) and Sun ($m=2.0 \times 10^{30}$ kg, $r=1.5 \times 10^{11}$ m) are on opposite sides of the Earth ($m=6.0 \times 10^{24}$ kg).

1. a. Calculate the force exerted on the Earth by the Moon:
2. b. Calculate the force exerted on the Earth by the Sun:
3. c. Calculate the Net Force on the Earth during the eclipse:

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2. Brass Ball on the Moon

A solid brass ball has a mass of 2.10 kilograms. The Moon has a radius of $1.74 \times 10^6$ meters and a mass of $7.35 \times 10^{22}$ kilograms.

1. a. Determine the acceleration due to gravity on the surface of the moon:
2. b. Determine the mass of the ball on the surface of the moon:
3. c. Determine the weight of the ball on the surface of the Earth:
4. d. Determine the weight of the ball on the surface of the moon:

VII. Complex Derivations

3. Earth Satellite Variable Derivation

A satellite of mass $m$ is in a circular orbit around the Earth (mass $M_e$) at a radius $a$ from the center of the Earth.

1. a. Write the equation for the gravitational force:
2. b. Write the equation for the acceleration of the satellite:
3. c. Is the acceleration at radius $a$ greater or less than surface $g$?
4. d. How much work is done by gravity to keep it in orbit?