When the planet Jupiter is at a distance of $824.7$ million kilometers from the Earth,its angular diameter is measured to be $35.72''$ of arc. Calculate the diameter of Jupiter.

$(a)$ How many astronomical units $(AU)$ make $1$ parsec?
$(b)$ Consider a sun-like star at a distance of $2$ parsecs. When it is seen through a telescope with $100$ magnification,what should be the angular size of the star? The Sun appears to be $(\frac{1}{2})^{\circ}$ from the Earth. Due to atmospheric fluctuations,the eye cannot resolve objects smaller than $1$ arc minute.
$(c)$ Mars has approximately half of the Earth's diameter. When it is closest to the Earth,it is at about $\frac{1}{2} AU$ from the Earth. Calculate what size it will appear when seen through the same telescope.

What is the uncertainty in time measured by a cesium atomic clock?

$A$ man wishes to estimate the distance of a nearby tower from him. He stands at a point $A$ in front of the tower $C$ and spots a very distant object $O$ in line with $AC$. He then walks perpendicular to $AC$ up to $B$,a distance of $100 \; m$,and looks at $O$ and $C$ again. Since $O$ is very distant,the direction $BO$ is practically the same as $AO$; but he finds the line of sight of $C$ shifted from the original line of sight by an angle $\theta = 40^{\circ}$ ($\theta$ is known as 'parallax'). Estimate the distance (in $m$) of the tower $C$ from his original position $A$.

Explain how time was measured in ancient times. Also,write a note on the cesium clock (atomic clock).

Which of the following time measuring devices is most precise?
$(A)$ $A$ wall clock
$(B)$ $A$ stop watch
$(C)$ $A$ digital watch
$(D)$ An atomic clock
Give reason for your answer.