13.7 Einstein's Theory of Gravity

Consider the density required to make Earth a black hole compared to that required for the Sun. What conclusion can you draw from this comparison about what would be required to create a black hole? Would you expect the Universe to have many black holes with small mass?

The principle of equivalence states that all experiments done in a lab in a uniform gravitational field cannot be distinguished from those done in a lab that is not in a gravitational field but is uniformly accelerating. For the latter case, consider what happens to a laser beam at some height shot perfectly horizontally to the floor, across the accelerating lab. (View this from a nonaccelerating…

What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?

A neutron star is a cold, collapsed star with nuclear density. A particular neutron star has a mass twice that of our Sun with a radius of 12.0 km. (a) What would be the weight of a 100-kg astronaut on standing on its surface? (b) What does this tell us about landing on a neutron star?

How far from the center of the Sun would the net gravitational force of Earth and the Sun on a spaceship be zero?

Suppose you can communicate with the inhabitants of a planet in another solar system. They tell you that on their planet, whose diameter and mass are $5.0 \times 10^{3} \text{km}$ and $3.6 \times 10^{23} \text{kg}$, respectively, the record for the high jump is 2.0 m. Given that this record is close to 2.4 m on Earth, what would you conclude about your extraterrestrial friends’ jumping ability?

A body of mass 100 kg is weighed at the North Pole and at the equator with a spring scale. What is the scale reading at these two points? Assume that $g = 9.83 \text{m}/\text{s}^{2}$ at the pole.

Consider the previous problem and include the fact that Earth has an orbital speed about the Sun of 29.8 km/s. (a) What speed relative to Earth would be needed and in what direction should you leave Earth? (b) What will be the shape of the trajectory?

An asteroid has speed 15.5 km/s when it is located 2.00 AU from the sun. At its closest approach, it is 0.400 AU from the Sun. What is its speed at that point?

A satellite of mass 1000 kg is in circular orbit about Earth. The radius of the orbit of the satellite is equal to two times the radius of Earth. (a) How far away is the satellite? (b) Find the kinetic, potential, and total energies of the satellite.

(a) Given the asteroid Vesta which has a diameter of 520 km and mass of $2.67 \times 10^{20} \text{kg}$, what would be the orbital period for a space probe in a circular orbit of 10.0 km from its surface? (b) Why is this calculation marginally useful at best?

(a) Using the information in the previous problem, what velocity do you need to escape the Milky Way galaxy from our present position? (b) Would you need to accelerate a spaceship to this speed relative to Earth?

Show that for eccentricity equal to one in the reference for conic sections, the path is a parabola. Do this by substituting Cartesian coordinates, x and y, for the polar coordinates, r and $\theta$, and showing that it has the general form for a parabola, $x = a y^{2} + b y + c$.

Given the perihelion distance, p, and aphelion distance, q, for an elliptical orbit, show that the velocity at perihelion, $v_{p}$, is given by $v_{p} = \sqrt{\frac{2 G M_{\text{Sun}}}{\left(q + p\right)} \frac{q}{p}}$. (Hint: Use conservation of angular momentum to relate $v_{p}$ and $v_{q}$, and then substitute into the conservation of energy equation.)

A tunnel is dug through the center of a perfectly spherical and airless planet of radius R. Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.

Show that the areal velocity for a circular orbit of radius r about a mass M is $\frac{\Delta A}{\Delta t} = \frac{1}{2} \sqrt{G M r}$. Does your expression give the correct value for Earth’s areal velocity about the Sun?

Show that for small changes in height h, such that $h < < \text{R}_{\text{E}}$, the reference reduces to the expression $\Delta U = m \text{g} h$.

(a) Show that tidal force on a small object of mass m, defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitation at a distance R from M, is given by $F_{\text{tidal}} = \frac{2 G M m}{R^{3}} \Delta r$ where $\Delta r$ is the distance between the near and far side and $\Delta r < < R$. (b) Assume you are falling feet first into the black hole at the center of our galaxy…