Ten sense-making tasks on universal gravitation, gravitational fields, and orbits. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Each case shows two uniform spheres far from any other objects. The masses and the center-to-center distances are given.
Rank the cases by the magnitude of the gravitational force the spheres exert on each other, from greatest to least. If any are equal, show the tie. If the ranking cannot be determined, say so.
Greatest 1 ____ 2 ____ 3 ____ 4 ____ 5 ____ 6 ____ Least
Explain your reasoning. Then answer: in case E, is the force on the 400 kg sphere larger than, smaller than, or equal to the force on the 100 kg sphere?
Back to topAstronauts on the International Space Station (ISS) float around the cabin. The ISS orbits about 400 km above Earth's surface. Three students explain why the astronauts float.
Ava: "There's no gravity in space. Once you're above the atmosphere, Earth can't pull on you anymore."
Ben: "Gravity up there is almost as strong as on the ground. The astronauts and the station are both falling toward Earth with the same acceleration, so the floor doesn't push on them."
Cam: "Gravity is still there, but the station is moving so fast that an outward centrifugal force cancels it. The net force on the astronauts is zero."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement. Estimate the gravitational field strength at the ISS (Earth's radius is 6400 km and $g = 10\ \text{m/s}^2$ at the surface).
Back to topA satellite moves counterclockwise in a circular orbit at constant speed. Its rockets are off. A student draws this free-body diagram for the satellite at the moment shown. The green dashed arrow is the satellite's velocity (not a force).
What, if anything, is wrong with this free-body diagram? If something is wrong, explain the error and describe the correct diagram. If nothing is wrong, explain why it is correct.
Back to topOn Earth's surface, $g = 10\ \text{m/s}^2$. For each planet described below (all are uniform spheres), choose what the gravitational field strength at its surface is:
A. greater than $10\ \text{m/s}^2$ B. equal to $10\ \text{m/s}^2$ C. less than $10\ \text{m/s}^2$ D. cannot be determined
| Planet | Choice |
|---|---|
| (a) Twice Earth's mass and twice Earth's radius | |
| (b) Four times Earth's mass and twice Earth's radius | |
| (c) Same mass as Earth, half Earth's radius | |
| (d) Twice Earth's mass, radius not given | |
| (e) Earth itself, but measured by dropping a rock with twice the mass | |
| (f) Same density as Earth, twice Earth's radius |
Explain the rule you used. Give the value of $g$ wherever you can.
Back to topEarth's mass is 81 times the Moon's mass. Consider only the gravitational interaction between them.
For each quantity, state whether it is greater for Earth, greater for the Moon, or the same for both. If it's different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Magnitude of the gravitational force exerted on each body by the other | |
| (b) Magnitude of each body's acceleration caused by that force | |
| (c) Gravitational field strength each body creates at the other body's location |
Explain each answer. If the Moon were moved to twice its present distance, by what factor would the force in (a) change?
Back to topTwo satellites move in circular orbits around Earth. Orbit 2 has four times the radius of Orbit 1. Earth's mass is $6.0\times10^{24}\ \text{kg}$.
The equation below was written by a student solving a problem about one object:
$$\frac{(6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2)(6.0\times10^{24}\ \text{kg})(500\ \text{kg})}{(2.0\times10^{7}\ \text{m})^2} = \frac{(500\ \text{kg})\,v^2}{2.0\times10^{7}\ \text{m}}$$
A 100 kg space probe measures the gravitational force a planet exerts on it at several distances from the planet's center. The planet's radius is $2.0\times10^{6}\ \text{m}$.
A satellite weighs 2000 N on Earth's surface. It is placed in orbit at an altitude of $2R_E$ above Earth's surface. Find the gravitational force on it in orbit.
Student solution:
Gravity follows an inverse-square law: $F \propto 1/r^2$.
The satellite is $2R_E$ away, so $r$ is 2 times bigger than on the surface.
$F = \dfrac{2000\ \text{N}}{2^2} = 500\ \text{N}$
Identify the error in the solution. Explain the physics of why it's an error, then find the correct force.
Back to topA 1000 kg probe is launched straight up and reaches a maximum height of $3.2\times10^{6}\ \text{m}$ above Earth's surface (half of Earth's radius).
Jordan writes: $\Delta U_g = mgh = (1000\ \text{kg})(10\ \text{m/s}^2)(3.2\times10^{6}\ \text{m}) = 3.2\times10^{10}\ \text{J}$, so the probe–Earth system gained $3.2\times10^{10}\ \text{J}$ of gravitational potential energy.