Three objects of equal mass m₁ = m₂ = m 3 = 1.75 kg are located on the vertices of an equilateral triangle of length d 3.05 m. A fourth mass m at the center of the triangle. What is the net gravitational force on the mass m₁ that is at the origin? Express your answer in vector form. = = 15.5 kg is F net (56.28-10-11)+(32.49-10-11) j Review the vector form of the gravitational law. What is the direction of the unit vector for each force acting on the mass located at the origin? N m3 m4 m₁ d m2 x

Suppose the gravitational acceleration at the surface of a certain moon A of Jupiter is 2 m/s2. Moon B has twice the mass and twice the radius of moon A. What is the gravitational acceleration at its surface? Neglect the gravitational acceleration due to Jupiter, (a) 8 m/s2 (b) 4 m/s2 (c) 2 m/s2 (d) 1 m/s2 (e) 0.5 m/s2

The Sun has a mass of approximately 1.99 1030 kg. a. Given that the Earth is on average about 1.50 1011 m from the Sun, what is the magnitude of the Suns gravitational field at this distance? b. Sketch the magnitude of the gravitational field due to the Sun as a function of distance from the Sun. Indicate the Earths position on your graph. Assume the radius of the Sun is 7.00 108 m and begin the graph there. c. Given that the mass of the Earth is 5.97 1024 kg, what is the magnitude of the gravitational force on the Earth due to the Sun?

Two black holes (the remains of exploded stars), separated by a distance of 10.0 AU (1 AU = 1.50 1011 m), attract one another with a gravitational force of 8.90 1025 N. The combined mass of the two black holes is 4.00 1030 kg. What is the mass of each black hole?

(a) Given that the period of the Moons orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84 108 m, use Equation 13.11 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large?

Model the Moons orbit around the Earth as an ellipse with the Earth at one focus. The Moons farthest distance (apogee) from the center of the Earth is rA = 4.05 108 m, and its closest distance (perigee) is rP = 3.63 108 m. a. Calculate the semimajor axis of the Moons orbit. b. How far is the Earth from the center of the Moons elliptical orbit? c. Use a scale such as 1 cm 108 m to sketch the EarthMoon system at apogee and at perigee and the Moons orbit. (The semiminor axis of the Moons orbit is roughly b = 3.84 108 m.)

For many years, astronomer Percival Lowell searched for a Planet X that might explain some of the perturbations observed in the orbit of Uranus. These perturbations were later explained when the masses of the outer planets and planetoids, particularly Neptune, became better measured (Voyager 2). At the time, however, Lowell had proposed the existence of a Planet X that orbited the Sun with a mean distance of 43 AU. With what period would this Planet X orbit the Sun?

Sphere A with mass 80 kg is located at the origin of an xy coordinate system; sphere B with mass 60 kg is located at coordinates (0.25 m, 0); sphere C with mass 0.20 kg is located in the first quadrant 0.20 m from A and 0.15 m from B. In unit-vector notation, what is the gravitational force on Cdue to AandB?

The diagram below shows three masses at the corners of a square of sides d = 1.10 m. Here, m, = m, = m and m, = 3.10m where m = 9.00 kg. m, m3 m1 (a) What is the magnitude of the gravitational field at the center of the square due to these three masses? 1.39e-9 What is the field at the center due to the two equal masses? Did you consider the symmetry of the situation? m/s2 (b) Suppose the two masses m, and m, are not equal. What value of m, will produce a gravitational field at the center of the square directed vertically down? kg

Two spheres of mass  M1 = 710 kg  and  M2 = 370 kg  are placed  4.60 m  apart. A particle of mass  m = 16.0 kg  is now placed midway between the two spheres. (a) What is the net gravitational force on the particle due to the two spheres? N toward the sphere of   M1, M2? (b) At what position between the two spheres should the particle be placed so that the net gravitational force on the particle is zero? m from the sphere of mass  M1