Chapter 13: Newton’s Theory of Gravity

The International Space Station orbits $300km$above the surface of the earth. What is the gravitational force on a $1.0kg$sphere inside the International Space Station?

The two stars in a binary star system have masses $2.0\times {10}^{30}kg$and $6.0\times {10}^{30}kg$. They are separated by $2.0\times {10}^{12}m$. What are

a. The system’s rotation period, in years?

b. The speed of each star?

A $4000kg$lunar lander is in orbit $50km$above the surface of the moon. It needs to move out to a $300km$high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?

The $75,000kg$space shuttle used to fly in a $250km$high circular orbit. It needed to reach a$610km$ high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?

How much energy would be required to move the earth into a circular orbit with a radius $1.0km$ larger than its current radius?

NASA would like to place a satellite in orbit around the moon such that the satellite always remains in the same position over the lunar surface. What is the satellite’s altitude?

In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days.
a. What was the satellite’s orbital speed around the comet? (Both the comet and the satellite are orbiting the sun at a much higher speed.)

b. What is the mass of the comet?
c. The lander was pushed from the satellite, toward the comet, at a speed of 70 cm/s, and it then fell—taking about 7 hours—to the surface. What was its landing speed? The comet’s shape is
irregular, but on average it has a diameter of 3.6 km.

A satellite orbiting the earth is directly over a point on the equator at 12:00 midnight every two days. It is not over that point at any time in between. What is the radius of the satellite’s orbit?

FIGURE P13.57 shows two planets of mass m orbiting a star of mass M. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. Find an exact expression for the orbital period T. Hint: Each planet feels two forces.

Figure 13.17 showed a graph of log T versus log r for the planetary data given in Table 13.2. Such a graph is called a log-log graph. The scales in Figure 13.17 are logarithmic, not linear, meaning
that each division along the axis corresponds to a factor of 10 increase in the value. Strictly speaking, the “correct” labels on the y-axis should be 7, 8, 9, and 10 because these are the logarithms of 10⁷...... 10¹⁰.
a. Consider two quantities u and v that are related by the expression v_p = Cu^q, where C is a constant. The exponents p and q are not necessarily integers. Define x = log u and y = log v. Find
an expression for y in terms of x.
b. What shape will a graph of y versus x have? Explain.
c. What slope will a graph of y versus x have? Explain.
d. Use the experimentally determined “best-fit” line in Figure 13.17 to find the mass of the sun.