I. Gravity Worksheet

A. Defined - Gravity is the of between any two objects in the universe.

$G$ – Universal Gravitational constant $= 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$

$m_1$ and $m_2$ – masses of the two attracting objects

r - The distance measured between the of the two objects.

B. Relationships

State the relationship between Gravitational Force (F) and distance (r): . What is the plot shape for this relationship?

If the distance between two objects is tripled, the gravitational force between them will change by a factor of

State the relationship between Gravitational Force (F) and mass (m): . What is the plot shape for this relationship?

Example 1: Two objects, with masses $m_1$ and $m_2$, are originally a distance r apart. The magnitude of the gravitational force between them is $F$. The masses are changed to $2m_1$ and $2m_2$, and the distance is changed to $4r$. What is the magnitude of the new gravitational force in terms of $F$?

Quick Check

3 questions on I. Defined & Relationships — pick an answer to check yourself.

1. If the distance between the centers of two objects is tripled, the gravitational force between them becomes…

A) F/3B) 3FC) F/9D) 9F

2. Two 100 kg spheres have centers 2.0 m apart. What is the gravitational force between them? (G = 6.67 × 10⁻¹¹ N·m²/kg²)

A) 3.3 × 10⁻⁷ NB) 1.7 × 10⁻⁷ NC) 6.7 × 10⁻⁷ ND) 1.7 × 10⁻⁹ N

3. Two masses feel gravitational force F. Both masses are doubled and the distance is made 4 times larger. What is the new force?

A) FB) F/2C) 4FD) F/4

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II. Satellite & Weight

Example 2: Satellite A and Satellite B have the same mass and are orbiting the same planet. If the gravitational force on satellite A is 160,000 N and it is at a distance $R$ from the center, what is the gravitational force on satellite B if it is at a distance of $2R$?

Example 3: A rocket weighs 18,000 N at the Earth's surface. If the rocket rises to a height where its total distance from the center of the Earth is 3 Earth radii ($3R_e$), calculate its weight at that new location.

When an object is on the Earth, the net gravitational force on that object is defined as its weight ($F_g = mg$).

C. Weight – The net gravitational force exerted on an object by a celestial body.

Quick Check

3 questions on II. Satellite & Weight — pick an answer to check yourself.

1. The gravitational force on a satellite at distance R from a planet's center is 160,000 N. What is the force on an identical satellite at 2R?

A) 40,000 NB) 80,000 NC) 20,000 ND) 640,000 N

2. A rocket weighs 18,000 N on Earth's surface. What does it weigh at a distance of 3 Earth radii from Earth's center?

A) 6,000 NB) 2,000 NC) 1,125 ND) 162,000 N

3. An astronaut moves from Earth's surface to 2 Earth radii from Earth's center. What happens to her mass and weight?

A) Both drop to one-fourthB) Mass halves, weight drops to one-fourthC) Mass stays the same, weight drops to one-fourthD) Mass stays the same, weight halves

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III. Surface Gravity

Variations in Gravity Near the Earth's Surface:

New York - 9.803 m/s² / Denver - 9.796 m/s² / North Pole - 9.832 m/s²

Based on the formula for gravitational acceleration, what physical factor explains the differences in $g$ at these locations?

Simplify the expression above to find the acceleration due to gravity $g$ on a planet's surface: $g = $

In this formula, $M_e$ represents the of the planet.

Example 4: Mars has a mass approximately 1/10 that of Earth and a diameter approximately 1/2 that of Earth. Based on this data, calculate the acceleration of a falling body near the surface of Mars in terms of $g$.

Giancoli p. 131 #28: A 2100-kg spacecraft is in a circular orbit at a distance of 2.0 Earth radii from the Earth's center. Calculate the magnitude of the force of gravity exerted on the spacecraft by the Earth.

Giancoli p. 131 #32: A hypothetical planet has a radius 1.5 times that of Earth, but has exactly the same mass as Earth. What is the value of the acceleration due to gravity on the surface of this hypothetical planet?

Gravitational Field Explorer

Compare surface g on Earth, the Moon, Mars, and Jupiter, then drag the test mass to see g = Gm/r² fall off with distance.

Gravitational Fields Notes

Quick Check

3 questions on III. Surface Gravity — pick an answer to check yourself.

1. A planet has the same mass as Earth but 1.5 times Earth's radius. What is g on its surface? (Earth's g = 9.8 m/s²)

A) 6.5 m/s²B) 22 m/s²C) 4.4 m/s²D) 9.8 m/s²

2. Mars has about 1/10 of Earth's mass and 1/2 of Earth's diameter. Its surface gravity is about…

A) 0.05gB) 0.2gC) 2.5gD) 0.4g

3. g is 9.803 m/s² in New York but only 9.796 m/s² in Denver. What best explains the difference?

A) Denver is at higher elevation, so it is farther from Earth's center (larger r) and g is smallerB) Denver has thinner air pushing down on objectsC) Objects in Denver have less massD) Denver is closer to the equator, so Earth's mass there is smaller

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IV. Orbital Proofs

Orbital Mechanics Simulator

Rank the gravitational forces from greatest to smallest.

Example 5: Use the principles of circular motion and universal gravitation to prove that the orbital velocity of a satellite is independent of its own mass.

a) From your derivation, identify which two physical quantities determine the orbital velocity $v$ and the orbital period $T$:

b) How does decreasing the orbital radius affect the orbital velocity $v$ and the orbital period $T$ ?

Quick Check

3 questions on IV. Orbital Proofs — pick an answer to check yourself.

1. A satellite orbits a planet with GM = 4.0 × 10¹⁴ m³/s² at r = 1.0 × 10⁸ m. What is its orbital speed?

A) 4.0 × 10⁶ m/sB) 2.0 × 10⁷ m/sC) 2.0 × 10³ m/sD) 1.0 × 10³ m/s

2. Satellite B has twice the mass of satellite A, and both orbit Earth at the same radius. Compared with A, B's orbital speed is…

A) The sameB) Twice as largeC) Half as largeD) √2 times as large

3. If a satellite moves to a smaller circular orbit, what happens to its orbital speed v and period T?

A) v decreases and T increasesB) v increases and T decreasesC) Both increaseD) Both stay the same

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V. Gravity Questions

VI. Extended Analysis
(Free Response)

1. Lunar Eclipse Net Force: During a total lunar eclipse, the Moon, Earth, and Sun are aligned in a straight line, with the Earth positioned directly between the Moon and the Sun. Use the following data to solve:
Mass of the Moon ($m_m$) = $7.4 \times 10^{22} \text{ kg}$
Mass of the Earth ($M_e$) = $6.0 \times 10^{24} \text{ kg}$
Mass of the Sun ($M_s$) = $2.0 \times 10^{30} \text{ kg}$
Earth-Moon distance ($r_{em}$) = $3.8 \times 10^8 \text{ m}$
Earth-Sun distance ($r_{es}$) = $1.5 \times 10^{11} \text{ m}$

1. Calculate the magnitude of the gravitational force exerted on the Earth by the Moon.
2. Calculate the magnitude of the gravitational force exerted on the Earth by the Sun.
3. Calculate the magnitude and direction of the net gravitational force exerted on the Earth by both the Moon and the Sun during this alignment.

2. 2.10-kg brass ball transported to Moon: A solid brass ball is manufactured with a mass of 2.10 kilograms. It is transported to the surface of the Moon.
Radius of the Moon = $1.74 \times 10^6$ meters
Mass of the Moon = $7.35 \times 10^{22}$ kilograms

1. Calculate the acceleration due to gravity on the lunar surface.
2. Compare the mass of the ball on Earth to its mass on the Moon. Explain any difference.
3. Calculate the weight of the ball when it is on the surface of the Earth.
4. Calculate the weight of the ball when it is on the surface of the Moon.

3. Earth Satellite Variable Derivation: A satellite of mass $m$ is placed in a stable circular orbit around the Earth. The Earth has a mass $M_e$ and a radius $R_e$. The satellite orbits at a distance $a$ measured from the center of the Earth. Express all algebraic answers strictly in terms of $a$, $m$, $M_e$, $R_e$, and $G$.

1. Derive the expression for the gravitational force acting on the satellite.
2. Derive the expression for the centripetal acceleration of the satellite.
3. Determine if the acceleration of the satellite in this orbit is greater than, less than, or equal to the acceleration $g$ at the surface of the Earth. Justify your answer.
4. Derive the expression for the tangential orbital velocity of the satellite.
5. How much work is done by the gravitational force on the satellite during one complete orbit? Justify your answer.
6. Derive the expression for the orbital period $T$ of the satellite.

Quick Check

3 questions on VI. Extended Analysis — pick an answer to check yourself.

1. A satellite moves in a circular orbit. How much work does gravity do on it during one complete orbit?

A) Zero, because gravity is always perpendicular to its motionB) Positive, because gravity pulls on it the whole timeC) Negative, because gravity pulls it inwardD) It depends on the satellite's mass

2. A 2.10 kg brass ball is taken to the Moon, where g ≈ 1.6 m/s². What is its weight there?

A) 2.1 NB) 21 NC) 3.4 ND) 1.3 N

3. A satellite orbits at distance a from Earth's center, where a is greater than Earth's radius Re. How does its acceleration compare with g at Earth's surface?

A) Greater than gB) Less than gC) Equal to gD) Zero, since it is weightless

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VII. Case Studies

4. Jupiter Satellite: A research satellite is in a stable circular orbit around Jupiter at a radius $R$. Jupiter has a mass $M_J = 1.90 \times 10^{27} \text{ kg}$ and a mean radius $R_J = 7.14 \times 10^7 \text{ m}$.

1. Starting from first principles ($F_g = F_c$), derive an expression for the orbital velocity of the satellite at a radius $R$ from Jupiter's center.
2. If the satellite were moved to a higher orbit with a larger radius, describe how its orbital velocity would change. Provide a physical explanation.
3. Derive an expression for the orbital period $T$ of the satellite.
4. Calculate the required orbital radius $R$ for a satellite that must have a period of exactly $3.55 \times 10^4 \text{ s}$.

5. Sojourner Rover Case Study: The Pathfinder spacecraft successfully landed on the Martian surface on July 4, 1997. It deployed a small rover named Sojourner to explore the landing site. Use the planetary and rover data provided below to complete the following analysis.

Mars Planetary Data: Radius = $0.53 R_e$, Mass = $0.11 M_e$.
Sojourner Rover Data:
Mass = $11.5 \text{ kg}$
Wheel diameter = $0.13 \text{ m}$
Stored Energy in Battery = $5.4 \times 10^5 \text{ J}$
Power consumption for driving = $10 \text{ W}$
Constant driving speed = $6.7 \times 10^{-3} \text{ m/s}$

1. Calculate the acceleration due to gravity at the surface of Mars expressed as a fraction of $g$ (Earth's surface gravity).
2. Calculate the weight of the Sojourner rover when it is on the surface of Mars.
3. To leave the lander, Sojourner must roll down a ramp inclined at 20° to the horizontal. Calculate the minimum normal force the ramp must provide to support the rover.
4. What is the magnitude of the net force acting on Sojourner as it travels across a level Martian surface at a constant velocity? Justify your answer.
5. Calculate the maximum total distance Sojourner can travel on a horizontal surface using only its stored battery energy.
6. If 0.010% of the driving power is used to overcome atmospheric drag on Mars, calculate the magnitude of the drag force acting on the rover.

Check Your Understanding

5 quick questions — pick an answer to see if you’re right.

1. Two masses m₁ and m₂ attract with force F at distance r. Both masses are doubled and the distance is changed to 4r. What is the new force?

A) F B) F/2 C) F/4 D) 4F

2. A rocket weighs 18,000 N at Earth's surface. What is its weight at 3 Earth radii from Earth's center?

A) 2,000 N B) 6,000 N C) 162,000 N D) 4,500 N

3. Mars has about 1/10 Earth's mass and 1/2 Earth's diameter. What is the gravitational acceleration at Mars's surface in terms of g?

A) 0.2g B) 0.05g C) 2.5g D) 0.4g

4. For a satellite in a circular orbit, which quantities determine its orbital speed?

A) The satellite's mass and the radius B) The central body's mass and the orbital radius C) Only the satellite's mass D) The satellite's mass and the central body's mass

5. How much work does gravity do on a satellite during one complete circular orbit?

A) A positive amount B) A negative amount C) Zero D) It depends on the satellite's mass

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