Two equal-mass stars maintain a constant distance apart of $8.0 \times 10^{11} \mathrm{~m}$ and revolve about a point midway between them at a rate of one revolution every $12.6 \mathrm{yr}$. $(a)$ Why don't the two stars crash into one another due to the gravitational force between them? $(b)$ What must be the mass of each star?

Binary star: Two equal-mass stars maintain a constant distance $D$ apart (Fig. 6-34) of $8.0 \times 10^{11} \mathrm{~m}$ and revolve about a point midway between them at a rate of one revolution every $14.2 \mathrm{yr}$. (a) Why don't the two stars crash into one another due to the gravitational force between them?

In a double-star system, two stars of mass $3.0 \times 10^{30} \text{~kg}$ each rotate about the system’s center of mass at radius $1.0 \times 10^{11} \text{~ m}$. If a meteoroid passes through the system’s center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to “infinity” from the two-star system?

(II) Take into account the Earth's rotational speed ($1\ \text{rev/day}$) and determine the necessary "escape velocity," with respect to the Earth, for a rocket to escape if fired from the Earth at the equator in a direction (c ) vertically upward.

(II) Take into account the Earth's rotational speed ($1\ \text{rev/day}$) and determine the necessary "escape velocity," with respect to the Earth, for a rocket to escape if fired from the Earth at the equator in a direction (b) westward

(II) Take into account the Earth's rotational speed ($1\ \text{rev/day}$) and determine the necessary "escape velocity," with respect to the Earth, for a rocket to escape if fired from the Earth at the equator in a direction (a) eastward

(III) The comet Hale-Bopp has an orbital period of $2400\:\text{years}$. (c ) What is the ratio of the speed at the closest point to the speed at the farthest point?

(III) The comet Hale-Bopp has an orbital period of $2400\:\text{years}$. (a) What is its mean distance from the Sun?

A block is released from rest and slides down an incline. The coefficient of sliding friction is 0.38, and the angle of inclination is 60.0 degree. Use energy considerations to find how fast the block is sliding after it has traveled a distance of 30.0 cm along the incline.

(II) Consider the track shown in Fig. 8-39.The section AB is one quadrant of a circle of radius $2.0\ \text{m}$ and is frictionless. B to C is a horizontal span $3.0\ \text{m}$ long with a coefficient of kinetic friction $\mu_k = 0.25$. The section CD under the spring is frictionless. A block of mass $1.0\ \text{kg}$ is released from rest A. After sliding on the track, it compresses the spring by $0.20\ \text{m}$. Determine: (d) the stiffness constant k for the spring.

A solid uniform sphere of radius a has a spherical cavity of radius a/2 centered at a point a/2 from the center of the sphere. Find the center of mass.

The nearest large galaxy to our Galaxy is about $2 \times 10^6$ ly away. If both galaxies have a mass of $3 \times 10^{41} \mathrm{~kg}$, with what gravitational force does each galaxy attract the other?

Estimate the mass of our galaxy (the Milky Way) if the Sun orbits the center of the galaxy with a period of 250 million years at a mean distance of $30,000 \text{c} \cdot \text{y}$. Express the mass in terms of multiples of the solar mass $M_{\text{S}}$. (Neglect the mass farther from the center than the Sun, and assume that the mass closer to the center than the Sun exerts the same force on the Sun, as would a point particle of the same mass located at the center of the galaxy.)

A satellite is in an elliptic orbit around the Earth (Fig. 8-51). Its speed at the perigee A is $8650\mathrm{~m/s}$. $(a)$ Use conservation of energy to determine its speed at B.