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Gravitation Questions

Questions

A spaceship is on a straight-line path between earth and moon. At what distance from the center of Earth is the net gravitational force on the ship is zero? (mass of Earth $M_E$, mass of moon $M_M$ and $d$ is the distance between center of Earth and the moon)

Turksat $2A$ is a communication satellite, which orbits on a geostationary orbit. The period of the satellites orbiting on the geostationary orbits is about $24$ hours. Therefore, from locations on the surface of the earth, Turksat 2A appears motionless. Calculate the radius of the geostationary orbit $R_G$. (Assume this orbit is a circular orbit. $M_E\sim \ 6\times {10}^{24}{\rm kg}$ and $G=6.6\times {10}^{-11}\ {\rm N.}\frac{{{\rm m}}^{{\rm 2}}}{{\rm k}{{\rm g}}^{{\rm 2}}}$) . 

The gravitational acceleration at the surface of a planet is $22.5\, {\rm m/}{{\rm s}}^{{\rm 2}}$. Find the acceleration at a height above the surface equal to the planet's radius.

The radius of a spherical planet is $R$ and its mass is $M$. At what distance above the planet's surface will the acceleration of gravity be exactly one tenth of its value at the surface?

You are on the home planet of the Klingon Empire. You drop a ball of mass $1.24\, {\rm kg}$ from a $334\, {\rm m}$ tower and it takes $6.54\, {\rm s}$ to reach the ground. The diameter of the Klingon home world is $1.55$ times the diameter of the earth. What is the mass of the Klingon home world?

What is the speed of a satellite of mass $ m=1.0 \times 10^{4}\, {\rm kg}$ on a circular orbit around the moon at a distance of ${\rm 4}{{\rm R}}_{moon}$ from the moon's surface? 

Astronauts put their spaceship into orbit about a planet. They find that the acceleration of gravity at their orbital altitude is half that at the planet's surface. How far above the planet's surface are they orbiting? Answer in terms of the radius of the planet.

Consider the ring-shaped body of radius $a$ and mass $M$ shown in the figure. A particle of mass $m$ is placed a distance $x$ from center of the ring, along a line through the center of the ring and perpendicular to it.

(a) Calculate the gravitational potential energy $U$ of this system. Take the potential energy to be zero when the two objects are very far apart. Recall that we can write $dU=-\frac{Gm}{r}dM$ where $r={\left(x^2+a^2\right)}^{\frac{1}{2}}$ as we integrate around the ring. 
(b) Show that your answer in part (a) reduces to the point masses result when $x$ is much larger than the radius $a$ of the ring. 
(c) Use $F_x=-dU/dx$ to find the magnitude and direction of the force on the particle.
(d) Show that your answer in part (c) reduces to the expected result when $x$ is much larger than $a$. What are the values of $U$ and $F_x$ when $x=0$ and why?


Gravitation
Category : Gravitation

MOST USEFUL FORMULA IN GRAVITY:

Newton's law of gravity:
\[F=-G\frac {m_1m_2}{r^2}\]
Universal gravitational constant:
\[G=6.67\times 10^{-11}\,\mathrm {N.m^{2}/kg^2}\]
Gravitational potential energy:
\[U(r)=-\frac{GmM}r\]


Number Of Questions : 8