Which of the following is the most precise device for measuring length:

$(a)$ a vernier callipers with $20$ divisions on the sliding scale

$(b)$ a screw gauge of pitch $1\; mm$ and $100$ divisions on the circular scale

$(c)$ an optical instrument that can measure length to within a wavelength of light?

In a screw gauge, there are $100$ divisions on the circular scale and the main scale moves by $0.5\,mm$ on a complete rotation of the circular scale. The zero of circular scale lies $6$ divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, $4$ linear scale divisions are clearly visible while $46^{\text {th }}$ division the circular scale coincide with the reference line. The diameter of the wire is $………..\times 10^{-2}\,mm$

A travelling microscope has $20$ divisions per $cm$ on the main scale while its Vernier scale has total $50$ divisions and $25$ Vernier scale divisions are equal to $24$ main scale divisions, what is the least count of the travelling microscope $……….\,cm$

Diagrams show readings of a screw gauge. figure $(i)$ shows the zero error reading when the screw gauge is closed, figure $(ii)$ the reading when the screw gauge is being used to measure the diameter of a ball-bearing. What is the diameter of the ball-bearing in $mm$? There are $50$ divisions on circular scale

The one division of main scale of vernier callipers reads $1\,mm$ and $10$ divisions of Vernier scale is equal to the $9$ divisions on main scale. When the two jaws of the instrument touch each other the $zero$ of the Vernier lies to the right of $zero$ of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the $zero$ of the Vernier scale lies in between $4.1\,cm$ and $4.2\,cm$ and $6^{\text {th }}$ Vernier division coincides with a main scale division. The diameter of the bob will be $………….10^{-2}\,cm$