Exercises: Newton's Law of Universal Gravitation

Two lab spheres of mass $7.0\ \text{kg}$ and $5.0\ \text{kg}$ sit on a table with their centers $0.50\ \text{m}$ apart. Use $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$ to calculate the magnitude of the gravitational force each sphere exerts on the other. State which sphere experiences the larger force and why.

Two planets exert a gravitational force $F$ on each other. Explain by what factor the force changes in each of these independent cases: (a) the distance between their centers is doubled; (b) the distance is cut to one third; (c) the mass of one planet is tripled while the distance stays the same.

The Moon has mass $7.35\times10^{22}\ \text{kg}$ and radius $1.74\times10^{6}\ \text{m}$. Use $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$ to calculate the gravitational field strength $g$ at the Moon's surface. Then find the weight on the Moon of an astronaut whose mass is $80\ \text{kg}$.

An astronaut of mass $70\ \text{kg}$ works on a space station orbiting $4.0\times10^{5}\ \text{m}$ above Earth's surface. Use $M_E = 5.97\times10^{24}\ \text{kg}$, $R_E = 6.37\times10^{6}\ \text{m}$, and $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. (a) Calculate the gravitational force on the astronaut at that altitude. (b) Calculate the astronaut's weight on Earth's surface. (c) The astronaut appears "weightless" on the station. Explain this in terms of your results.

A newly discovered planet has a mass 4 times Earth's mass and a radius 2 times Earth's radius. Take $g_{\text{Earth}} = 9.8\ \text{m/s}^2$ at Earth's surface. (a) Find the planet's surface gravitational field strength. (b) Find the weight on this planet of a student of mass $60\ \text{kg}$. (c) Explain whether the student's mass changes when moving to this planet.

A student measures the gravitational force between two small spheres at several separations (center to center) and records the data below.

(a) Predict the missing values, assuming the force follows Newton's law of universal gravitation. (b) Use $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$ to find the product $m_1 m_2$ of the two masses. (c) Explain how you could use a graph of $F$ versus $1/r^2$ to check that the data fit the law.

The Earth–Moon distance (center to center) is $3.84\times10^{8}\ \text{m}$. Use $M_E = 5.97\times10^{24}\ \text{kg}$ and $M_M = 7.35\times10^{22}\ \text{kg}$. A spacecraft lies on the line joining the two bodies. (a) Find the distance from Earth's center at which the net gravitational force on the spacecraft is zero. (b) Explain why the answer does not depend on the spacecraft's mass. (c) Explain whether the net force points toward Earth or toward the Moon if the spacecraft is slightly closer to Earth than this point.

Two identical spheres of mass $10\ \text{kg}$ are fixed at $(-0.30\ \text{m}, 0)$ and $(+0.30\ \text{m}, 0)$. A small sphere of mass $2.0\ \text{kg}$ is placed at $(0, 0.40\ \text{m})$. Use $G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2$. (a) Draw a free-body diagram for the small sphere, and find the magnitude of the force from each large sphere. (b) Determine the magnitude and direction of the net force on the small sphere. (c) Explain why the $x$-components cancel, and how the net force would change if the right-hand sphere had mass $20\ \text{kg}$.