Newton's Law of Gravity Practice Problems

A rock on Mars has a mass of 1200 kg. At midday, the rock is directly between the sun and Mars. i) Determine the force exerted on the rock by the sun. Take the distance from the rock to the sun's center as 2.28 × 10¹¹ m. ii) Contrast this force with the force on the rock from Mars (g_M = 3.72 m/s²).

A spherical mass used for short put has a mass m and diameter d. If two masses are separated by d/2 (from surface to surface) what will be the magnitude of gravitational force on the balls?

For the following configuration of masses, determine the magnitude and direction of the sum of the gravitational force that spheres B and D exert on sphere C. Take all spheres to have the same mass equal to 6.0kg.

Particles of mass 2m and 3m are placed on the y-axis. The 2m mass lies at the origin while 3m has a position y. A third particle is placed on a point on the y-axis where the sum of gravitational force on it from the other two particles is zero. Give the coordinate of the point in terms of y. 

To study Newton's law of gravity, a student fixes two metallic spheres each of mass 5.0 kg separated by 24.0cm. The student positions a third sphere of mass 0.50 kg at 5.0 cm from the midpoint of the line connecting the two spheres. The student releases the third sphere from rest. What is the acceleration magnitude and direction of the third sphere if it experiences only gravitational force from the other two spheres? Treat all masses like point masses.

Four celestial objects, each possessing a mass equivalent to the Earth, arrange themselves into a square configuration, with each side measuring 1.4 x 10¹¹ m. In order to maintain this structure and prevent a gravitational collapse toward the center, the square must uniformly rotate. What is the duration of one complete rotation period?

In a celestial system, two massive entities with masses of 3 x 10²⁸ kg and 7 x 10²⁸ kg are positioned 3.4 x 10¹¹ m apart from each other. Determine the speed of both entities in this system 

Consider a pair of balls (spherical) that have a collective mass of 211 kg. They are subject to a gravitational force of 6.25 x 10^-8 N, and their gravitational potential energy is ₋12.4 x 10^-8 J. Determine the individual mass of each ball in the given scenario.

A cylindrical object with a mass of 2.3 kg is located on a satellite that orbits Mars at an altitude of 400 km from its surface. Determine the gravitational force experienced by this cylinder. The mass of Mars is (6.42 × 10²³ kg) and its radius is  (3.34  × 10⁶ m).

Two satellites, named CosmoComm and SolarSurfer, each with a mass of 120 kg, are launched into space and positioned 48 meters apart from each other. At what distance from the Earth's center will the gravitational force between CosmoComm and SolarSurfer be equal to the gravitational force exerted by Earth on a single satellite?

How much gravitational attraction do an 80 kg astronaut and a 140 kg alien have on each other when they are seated 2.1 meters away from each other?

Two metallic spheres have a total mass of 210 kg. When the centers of these spheres are separated by 18 cm, they experience a gravitational pull of 9 x 10^-6 N. Determine the individual mass of each metallic sphere in this scenario.

Consider a system of clay balls of different masses placed at various positions in space, as shown in the figure. In the given scenario, calculate the net gravitational force on the 25 kg mass. Express your answer in unit vectors.

What is the percentage decrease in gravity strength at an altitude of 400 km above the Earth's surface? Given values are R_Earth=6.38×10⁶ m, G=6.67×10^-11 N.m²/kg², M_Earth=5.97×10²⁴ kg.

A rover operating on Mars measures changes in gravitational pull as it climbs a steep hill of height ∆z, where ∆z is much smaller than R_Mars (the radius of Mars). Use the binomial expansion

(1 ± x)ⁿ = 1 ± nx + (n(n-1)/2) x² ± ...

to derive the formula approximating the change in acceleration due to gravity in Mars $\Delta g_{\text{Mars }}$, where $g_{Mars}$ is the gravitational acceleration at Mars's surface.

As a Mars rover ascends a hill of height ∆z above the Martian surface, where ∆z is much smaller than $R_{Mars}$ (the radius of Mars), the value of the gravitational acceleration, $g$, is altered. The approximate change in $g$ can be expressed as

$\text{∆}{g}_{Mars}\approx -2g_{Mars}\frac{\text{∆}z}{R_{Mars}}$

where $g_{Mars}$ is the gravitational acceleration at Mars's surface. What is the meaning of the minus sign in this relation?

As a Mars rover climbs a hill of height ∆z above the Martian surface, where ∆z is much smaller than $R_{Mars}$ (the radius of Mars), the value of the gravitational acceleration, g, is altered. The change in gravitational acceleration $∆g_{Mars}$ at this height can be approximated by:

$∆g_{Mars}\approx -2g_{Mars}\ \frac{∆z}{R_{Mars}}$

where $g_{Mars}$ is the gravitational acceleration at Mars's surface. Use this equation to compute the effective value of g at 10 km above Mars’s surface. Compare this to the value of g using Newton's law of universal gravitation, $g=G\frac{m_{Mars}}{r^2}$, where r is the distance from the center of Mars to the location of the rover. Use: $g_{Mars}=3.71\ \frac{m}{s^2}$, $m_{Mars}=6.417\times10^{23}\mathrm{~kg}$, $R_{\text{Mars }}=3389.5\ \text{km}$.

A tall tree stands alone in the middle of a desert. A physics professor intends to measure its mass with a pendulum. He takes a bob of mass M and places the pendulum at a distance d = 200 m away from the tree. As a consequence, the bob makes an angle of Φ with the vertical due to the gravitational attraction between the bob and the tree. Given that the mass of the tree is 640000 kg, determine the angle Φ.

Archeologists place a pendulum on the surface of the Earth at a distance of D away from the center of a gigantic pyramid of mass M_p. The bob of the pendulum (having mass m) makes an angle of Φ with the vertical at the equilibrium due to the presence of the pyramid. Given that the radius of the Earth is R_E and the mass of the Earth is M_E, evaluate an expression of Φ in terms of M_p, D, R_E, and M_E.

An astronaut who can throw a discus to a maximum distance of 65.0 m on Earth participates in a space sports event on a planet that is the same size as Earth. There, given the same initial speed and angle, the maximum distance she can throw the discus is reduced to 40.0 m. Calculate the mass of this planet. Use $G=6.674\times 10^{-11\ }\frac{N.m^2}{kg^2}$ and radius of Earth $r_E=6.371\times 10^6\mathrm{~m}$.

Consider a celestial system where two stars, each of identical mass, orbit around their common center of mass which is located precisely at the midpoint of the line joining them. These stars are separated by 10 × 10¹¹ m. They complete one full orbit around their common center of mass every 12.6 years. Determine the mass of each star in this dual-star system.

Scientists are closely observing a group of three satellites situated in the void of space. They are arranged in a straight line, and their thrusters aren't working. There is nothing else in their close vicinity. The leftmost satellite is at a distance of 23 m from the one at the center, while the rightmost one is 43 m away from the one in the middle. Given that the leftmost satellite has a mass of 1700 kg and the net gravitational force on the satellite in the middle is zero, calculate the mass of the rightmost satellite.

Determine the individual masses of two spheres that attract each other with a gravitational force of 3.5×10⁻¹⁰ N when they are 0.28 m apart and their total mass is 5.00 kg.