Assignment

Identify the incorrect statement about a planet revolving around Sun.

Suppose a tunnel is made in earth which passes, through its centre. If an object is dropped from height $\frac{R}{2}$ from earth surface above the tunnel. If its speed at depth $\frac{R}{2}$ from earth surface in tunnel is $\sqrt{\frac{NGM}{12\mathrm{R}}}$, then value of $N$ is (Given $M$ is mass of earth and $R$ is radius of earth).

Six stars of equal masses $M$ are moving about the centre of mass of the system such that they are always on the vertices of a regular hexagon of side length a. If their time period of motion is $2\pi \sqrt{\frac{N\sqrt{3}{a}^3}{GM\left(5\sqrt{3}+N\right)}}$ then value of $N$ is ____.

A planet is revolving around the sun in an elliptical orbit. If the speed of planet at shown position $P$ ($M$:Mass of Sun) is $\sqrt{\frac{kGM}{2\mathrm{a}}}$, then value of $k$ is ______.

If height from the earth's surface, where weight of a man decreases by $10\%$ ($R$ is the radius of earth) is $\left(\sqrt{\frac{N}{g}}-1\right)\mathrm{R}$, then value of $N$ is _______.

A ring of mass $M$ and radius $R$ is fixed and a point mass $m$ is released from rest as shown in figure. If speed of the particle when it reaches the centre of the ring is $\sqrt{\left(k-\frac{k}{\sqrt{3}}\right)\frac{GM}{R}}$, then value of $k$ is ______.

A satellite of mass $m$ is moving around earth in an orbit of radius $2\mathrm{R}$. If amount of energy required to shift it in a orbit of radius $3\mathrm{R}$ is (Mass of the earth is $m_e$)  $\frac{G{m}_{e}m}{kR}$, then value of $k$ is ______.

If speed of planet at apogee in terms of eccentricity $\left(e\right)$ is $a$ is semimajor axis of ellipse) $\sqrt{\frac{kGM}{4\mathrm{a}}\left(\frac{1-e}{1+e}\right)}$, then value of $k$ is ______.