Problem 3
A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of 65 kg and the other a mass of 72 kg, and they start from rest 20.0 m apart. (a) Make a free- body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres. (b) If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They \(both\) have acceleration toward each other.) (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?
Problem 4
Two uniform spheres, each with mass \(M\) and radius \(R\), touch each other. What is the magnitude of their gravitational force of attraction?
Problem 7
A typical adult human has a mass of about 70 kg. (a) What force does a full moon exert on such a human when it is directly overhead with its center 378,000 km away? (b) Compare this force with the force exerted on the human by the earth.
Problem 8
An 8.00-kg point mass and a 12.0-kg point mass are held in place 50.0 cm apart. A particle of mass \(m\) is released from a point between the two masses 20.0 cm from the 8.00-kg mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.
Problem 9
A particle of mass 3\(m\) is located 1.00 m from a particle of mass \(m\). (a) Where should you put a third mass \(M\) so that the net gravitational force on \(M\) due to the two masses is exactly zero? (b) Is the equilibrium of \(M\) at this point stable or unstable (i) for points along the line connecting m and 3\(m\), and (ii) for points along the line passing through \(M\) and perpendicular to the line connecting \(m\) and 3\(m\)?
Problem 10
The point masses \(m\) and 2\(m\) lie along the x-axis, with \(m\) at the origin and 2\(m\) at \(x\) \(=\) \(L\). A third point mass \(M\) is moved along the \(x\)-axis. (a) At what point is the net gravitational force on \(M\) due to the other two masses equal to zero? (b) Sketch the \(x\)-component of the net force on \(M\) due to \(m\) and 2\(m\), taking quantities to the right as positive. Include the regions \(x < 0\), \(0 < x < L\), and \(x > L\). Be especially careful to show the behavior of the graph on either side of \(x = 0\) and \(x = L\).
Problem 11
At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 m/s\(^2\) if the acceleration due to gravity at the surface has magnitude 9.80 m/s\(^2\)?
Problem 12
The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venus?
Problem 13
Titania, the largest moon of the planet Uranus, has \(\frac{1}{8}\) the radius of the earth and \(\frac{1}{1700}\) the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)
Problem 14
Rhea, one of Saturn's moons, has a radius of 764 km and an acceleration due to gravity of 0.265 m/s\(^2\) at its surface. Calculate its mass and average density.
