Embibe Experts Solutions for Chapter: Gravitation, Exercise 1: Exercise 1

A certain planet completes one rotation about its axis in time $T.$ The weight of an object placed at the equator on the planet's surface is a fraction $f$ ($f$  is close to unity) of its weight recorded at a latitude of $60°.$ The density of the planet (assumed to be a uniform perfect sphere) is given by:

A planet revolves about the sun in elliptical orbit. The areal velocity $\left(\frac{\mathrm{d}A}{\mathrm{d}t}\right)$ of the planet is $4.0\times {10}^{16} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$. The least distance between planet and the sun is $2\times 10^{12} \mathrm{m}$. Then the maximum speed of the planet in $\mathrm{km} {\mathrm{s}}^{-1}$ is-

A comet (assumed to be in an elliptical orbit around the sun) is at a distance of $0.4 \mathrm{AU}$ from the sun at the perihelion. If the time period of the comet is $125$ years, what is the aphelion distance? $\mathrm{AU}$: astronomical unit.

A body is projected from the Earth's surface with a velocity equal to half the escape velocity from that point at an angle ${45}^{\mathrm{o}}$ with horizontal. The maximum height attained by the body above the Earth's surface is (nearly)

An astronaut whose height is $1.50 \mathrm{m}$ floats "feet down" in an orbiting space shuttle at ${\mathrm{a}}_{\mathrm{g}}$ distance $r=\sqrt[3]{6.67}\times {10}^6 \mathrm{m}$ away from the centre of Earth. The magnitude of difference between the gravitational acceleration at her feet and at her head is found to be $N\times {10}^{-5} \mathrm{m} {\mathrm{s}}^{-2}.$ What is the value of $N$ ? [mass of earth $=6\times {10}^{24} \mathrm{kg}$  and $G=6.67\times {10}^{-11} \mathrm{N} {\mathrm{m}}^2 {\mathrm{kg}}^{-2}]$

A hollow sphere (mass $\mathrm{m}$, radius $\mathrm{R}$ ) is put inside a hollow cone (mass $\mathrm{m}$, radius $3\mathrm{R}$ and semi-angle $30°$ ) as shown in the figure. The whole arrangement is rigidly fixed to the ground at the apex of the cone. Both the sphere and the cone are made of the same material $\left(\alpha ={10}^{-4}/{}^{\circ }\mathrm{C}\right)$. Considering that major changes in dimensions could happen only due to heating effects, if the increase in potential energy (in Joule) of the system if it's temperature is increased by $100°\mathrm{C}$ is $∆U$, what is the value of $10∆U$?

$\mathrm{m}=0.5 \mathrm{kg}$  

$\mathrm{R}=\frac{\sqrt{3}-1}{2} \mathrm{m}$

Two identical planets are orbiting around a star in elliptical paths having their semi-major axes in the ratio of $1: 9.$ If life of man on $1^{st}$ planet is $80$ years then what will be his life on $2^{nd}$ planet in terms of that planet's year? (Approximate the answer to the nearest integer)

Gravitational acceleration on the surface of a planet is $\frac{\sqrt{6}}{11} g,$ where $g$ is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $\frac{2}{3}$ times that of the earth. If the escape speed on the surface of the earth is taken to be $11km{s}^{-1},$ find the escape speed on the surface of the planet in $km{s}^{-1}.$