$A$ satellite of mass $m$ is at a distance of $a$ from a star of mass $M$. The speed of the satellite is $u$. Suppose the law of universal gravity is $F = -G \frac{Mm}{r^{2.1}}$ instead of $F = -G \frac{Mm}{r^2}$. Find the speed of the satellite when it is at a distance $b$ from the star.

Take the mean distance of the moon and the sun from the earth to be $0.4 \times 10^6 \, km$ and $150 \times 10^6 \, km$ respectively. Their masses are $8 \times 10^{22} \, kg$ and $2 \times 10^{30} \, kg$ respectively. The radius of the earth is $6400 \, km$. Let $\Delta F_1$ be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and $\Delta F_2$ be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then,the number closest to $\frac{\Delta F_1}{\Delta F_2}$ is

Mean solar day is the time interval between two successive noons when the sun passes through the zenith point (meridian). $A$ sidereal day is the time interval between two successive transits of a distant star through the zenith point (meridian). By drawing an appropriate diagram showing the earth's spin and orbital motion,show that the mean solar day is $4\,\text{min}$ longer than the sidereal day. In other words,distant stars would rise $4\,\text{min}$ early every successive day.

$A$ planet is moving in an elliptical orbit around the sun. The work done on the planet by the gravitational force of the sun:
$(i)$ is zero in no part of the motion.
(ii) is zero in some parts of the orbit.
(iii) is zero in one complete revolution.
(iv) is zero in any small part of the orbit.
Which of the following is true?

State whether the following statements are true or false:
$(a)$ The value of '$g$' is slightly less at the poles than at the equator.
$(b)$ The unit $\text{Newton/kilogram}$ $(N/kg)$ is also a unit of acceleration due to gravity.
$(c)$ The escape velocity $v_e$ for a stationary object on the surface of a planet is directly proportional to the mass and radius of the planet.

Which of the following astronomers first proposed that the Sun is static and the Earth revolves around the Sun?