Gravity Worksheet
A. Defined - of between any two objects in the universe
$$F = \frac{G m_1 m_2}{r^2}$$
$G - \text{Gravitational constant} = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$
$m_1 \text{ and } m_2 \text{ – masses attracting each other}$
r -
B. Relationships
Relationships – F/r Plot shape
Triple the distance causes the gravitational force to change by
F, m Plot shape
Ex 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?
Teacher Workspace: Deriving Ex 1
Module I Teacher Key
Claim: The magnitude of the new gravitational force is $F/4$.
Evidence: Substituting the variables into the formula: $F_{new} = G \frac{(2 m_1)(2 m_2)}{(4 r)^2} = \frac{4 G m_1 m_2}{16 r^2}$.
Reasoning: Gravitational force is directly proportional to the product of masses and inversely proportional to the square of the distance. While the mass product increases force by 4x, the distance factor decreases it by 16x ($4^2$). Result: $4/16 = 1/4$ the original force.
II. Satellite & Weight
Ex 2) If the gravitational force on satellite A is 160,000 N, what is the gravitational force on satellite B? Both satellites have the same m.
Ex 3) A rocket weighs 18,000 N at the Earth's surface. If the rocket rises to a height where its distance from the Earth is 3 Earth radii, its weight will be ...
When an object is on the Earth, the net gravitational force on that object is its weight.
C. Weight – Net gravitational force on an object
Teacher Workspace: Weight and Radius Analysis
Module II Teacher Key
Claim: The rocket's weight at 3 Earth radii is 2,000 N.
Evidence: $F_g \propto 1/r^2$. The distance increases from $1R_e$ to $3R_e$.
Reasoning: Weight is simply the pull of gravity. Since the distance from the center of mass has tripled, the force of attraction is divided by $3^2 = 9$. $18,000 \text{ N} / 9 = 2,000 \text{ N}$.
III. Surface Gravity
Gravity Near the Earth's Surface:
NY - 9.803 m/s² / Denver - 9.796 m/s² / North Pole - 9.832 m/s²
What explains the differences above?
$$(Weight) \text{ } F_g = mg = \frac{G m m_e}{r^2}$$
Simplify $g = $
$M_e - $
Ex 4) Mars has a mass 1/10 that of Earth and a diameter 1/2 that of Earth. The acceleration of a falling body near the surface of Mars is most nearly
Giancoli p. 131 #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.
Giancoli p. 131 #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?
Teacher Workspace: Deriving Planetary g
Module III Teacher Key
Claim: Mars surface gravity is $0.4g$.
Evidence: $g_{mars} = G \frac{(0.1 M_e)}{(0.5 R_e)^2} = \frac{0.1}{0.25} g_{earth} = 0.4 g$.
Reasoning: Surface gravity depends on the mass to radius-squared ratio. While Mars has 10x less mass, its smaller size places the surface closer to the center, strengthening gravity by 4x. The net result is 40% of Earth's gravity.
IV. Orbital Proofs
Ex 5) Prove that the orbital velocity of a satellite does not depend on its mass
Teacher Workspace: $F_g = F_c$ Derivation
a) From the equation, we see that orbital v and Period T are only determined by
b) The smaller the radius the the orbital velocity and period T
Module IV Teacher Key
Claim: The mass of the orbiting satellite has no effect on its required orbital velocity.
Reasoning: By setting Gravitational Force equal to Centripetal Force ($G \frac{M m_s}{r^2} = \frac{m_s v^2}{r}$), the satellite mass ($m_s$) cancels out algebraically. This proves that any object at a specific distance from a central body must travel at the same speed to maintain that orbit.
VI. Extended Analysis (Free Response)
1. Lunar Eclipse Net Force: During a lunar eclipse, the Moon, Earth, and Sun all lie on the same line, with the Earth between the Moon and the Sun. The Moon has a mass of $7.4 \times 10^{22} \text{ kg}$; the Earth has a mass of $6.0 \times 10^{24} \text{ kg}$; and the Sun has a mass of $2.0 \times 10^{30} \text{ kg}$. The separation between the Moon and the Earth is $3.8 \times 10^8 \text{ m}$; the separation between the Earth and the Sun is $1.5 \times 10^{11} \text{ m}$.
- Calculate the force exerted on the Earth by the Moon.
- Calculate the force exerted on the Earth by the Sun.
- Calculate the net force exerted on the Earth by the Moon and the Sun.
Teacher Workspace: Vector Calculations
2. 2.10-kg brass ball transported to 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.
- Calculate the acceleration due to gravity on the surface of the Moon.
- Determine the mass of the ball on the surface of the Earth and on the surface of the Moon.
- Determine the weight of the ball on the surface of the Earth.
- Determine the weight of the ball on the surface of the Moon.
Teacher Workspace: Mass vs Weight Proof
3. Earth Satellite Variable Derivation: A satellite of mass m is in a circular orbit around the Earth, which has mass Me and radius Re. The satellite is at a distance a from the center of the Earth. All algebraic answers should be in terms of a, m, Me, Re, and G.
- Write the equation for the gravitational force on the satellite.
- Write the equation for the acceleration of the satellite.
- Find the acceleration of the satellite if it is to stay on a circular orbit of radius a. Is this acceleration greater or less than the acceleration g at the surface of the Earth?
- Determine the velocity of the satellite as it stays in orbit.
- How much work is done by the gravitational force to keep the satellite in orbit?
- What is the orbital period of the satellite?
Teacher Workspace: Algebraic Derivations
Module VI Teacher Key
Claim (Work): The work done by gravity is zero Joules.
Evidence: Work is $W = F \cdot d \cdot \cos\theta$. In circular orbit, $F_g$ is perpendicular to displacement ($\theta = 90^\circ$).
Reasoning: Work is only done when a component of force acts along the direction of motion. Since gravity acts radially and the satellite moves tangentially, the dot product is zero. Energy is redirected, not added.
VII. Case Studies
4. Jupiter Satellite: A satellite of Jupiter has a circular orbit of radius R. Jupiter has a mass $M_J = 1.90 \times 10^{27} \text{ kg}$ and a radius $R_J = 7.14 \times 10^7 \text{ m}$.
- Derive an expression for the orbital velocity of the satellite at a radius R from the center of Jupiter.
- If the radius of the orbit increases, how would it change the orbital velocity of the satellite? Explain.
- Derive an expression for the orbital period of the satellite.
- Find the required orbital radius for a satellite to have a period of $3.55 \times 10^4 \text{ s}$.
Teacher Workspace: Jupiter Orbital Analysis
5. Sojourner Rover Case Study: The Pathfinder spacecraft landed on the Martian surface on July 4, 1997. It carried a small rover, Sojourner, which explored the Martian surface. Use the data on Mars below to answer the following.
Mars Data: Radius = $0.53 R_e$, Mass = $0.11 M_e$.
Sojourner Data: Mass = $11.5 \text{ kg}$, Wheel diameter = $0.13 \text{ m}$, Stored Energy = $5.4 \times 10^5 \text{ J}$, Power for driving = $10 \text{ W}$, Constant speed = $6.7 \times 10^{-3} \text{ m/s}$.
- Determine the acceleration due to gravity at the surface of Mars in terms of g, the acceleration due to gravity at the surface of Earth.
- Calculate Sojourner's weight on the surface of Mars.
- Assume that when leaving the Pathfinder spacecraft Sojourner rolls down a ramp inclined at 20° to the horizontal. The ramp must be lightweight but strong enough to support Sojourner. Calculate the minimum normal force that must be supplied by the ramp.
- What is the net force on Sojourner as it travels across the Martian surface at constant velocity? Justify your answer.
- Determine the maximum distance that Sojourner can travel on a horizontal Martian surface using its stored energy.
- Suppose that 0.010% of the power for driving is expended against atmospheric drag as Sojourner travels on the Martian surface. Calculate the magnitude of the drag force.
Teacher Workspace: Sojourner Detailed Proofs
Module VII Teacher Key
Claim (Sojourner Net Force): The net force is exactly zero Newtons.
Reasoning: According to Newton's First Law, an object traveling at a constant velocity is in dynamic equilibrium. This means all forces (driving force vs friction/drag) are perfectly balanced, resulting in zero net acceleration and zero net force.
Claim (Range): Sojourner can travel approximately 362 meters.
