Resnick & Halliday Solutions for Chapter: Gravitation, Exercise 1: Problems

In the figure shown below, a spherical hollow inside a lead sphere of radius $R=2.00 \mathrm{cm}\text{,}$ the surface of the hollow passes through the centre of the sphere and touches the right side of the sphere. The mass of the sphere before hollowing was $M=2.95 \mathrm{kg}$. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass $m=0.950 \mathrm{kg}$ that lies at a distance $d=9.00 \mathrm{cm}$ from the centre of the lead sphere, on the straight line connecting the centres of the spheres and of the hollow?

At what angular speed of rotation is the surface material on the equator of a neutron star on the verge of flying off the star, if the star is spherical with a radius of $18.0 \mathrm{km}$ and a mass of $7.72\times {10}^{24} \mathrm{kg}$?

Mountain pull. A large mountain can slightly affect the direction of down as determined by a plumb line. Assume that we can model a mountain as a sphere of radius $R=1.80 \mathrm{km}$ and density (mass per unit volume) $2.6\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$. Also assume that we hang a $0.50 \mathrm{m}$ plumb line at a distance of $3R$ from the sphere's centre and such that the sphere pulls horizontally on the lower end. How far would the lower end moves towards the sphere?

The Sun, which is $2.2\times {10}^{20} \mathrm{m}$ from the centre of the Milky Way galaxy, revolves around that centre once every $2.5\times {10}^8 \mathrm{years}$. Assuming each star in the galaxy has a mass equal to the Sun's mass of $2.0\times {10}^{30} \mathrm{kg},$ the stars are distributed uniformly in a sphere about the galactic centre and the Sun is at the edge of that sphere, estimate the number of stars in the galaxy.

Assume a planet is a uniform sphere of radius $R$ that (somehow) has a narrow radial tunnel through its centre. Also, assume we can position an apple anywhere along the tunnel or outside the sphere. Let $F_{\mathrm{R}}$ be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface is there a point where the magnitude is $\frac{1}{3}{F}_{\mathrm{R}}$, if we move the apple $\left(a\right)$ away from the planet and $\left(b\right)$ into the tunnel?

Two particles are separated by $19 \mathrm{m}$. If the mass of particle $1$ is $5.2 \mathrm{kg}$ and mass of particle $2$ is $4.2 \mathrm{kg}$

$\left(a\right)$ What is the gravitational potential energy of the two particle system? If you triple the separation between the particles, how much work is done $\left(b\right)$ by the gravitational force between the particles and $\left(c\right)$ by you?

A satellite orbits a planet of unknown mass in a circle of radius $3.0\times {10}^7 \mathrm{m}.$The magnitude of the gravitational force on the satellite from the planet is $F=80 \mathrm{N}$. $\left(a\right)$ What is the kinetic energy of the satellite in this orbit? $\left(b\right)$ What would $F$ be if the orbit radius was increased to $4.0\times {10}^7 \mathrm{m}$?

A $30 \mathrm{kg}$ satellite has a circular orbit with a period of $2.0 \mathrm{h}$ and a radius of $9.0\times {10}^6 \mathrm{m}$ around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is $6.0 \mathrm{m} {\mathrm{s}}^{-2}$, what is the radius of the planet?