The internal and external radii of a hollow cylinder are measured with the help of a vernier callipers. Their values are $(4.23 \pm 0.01) \, cm$ and $(3.87 \pm 0.01) \, cm,$ respectively. The thickness of the wall of the cylinder is

$A$ screw gauge of pitch $0.5\,mm$ is used to measure the diameter of a uniform wire of length $6.8\,cm$. The main scale reading is $1.5\,mm$ and the circular scale reading is $7$. The calculated curved surface area of the wire to appropriate significant figures is $......\,cm^2$. [Screw gauge has $50$ divisions on the circular scale]

$A$ student measured the diameter of a small steel ball using a screw gauge of least count $0.001 \, cm$. The main scale reading is $5 \, mm$ and the circular scale division coinciding with the reference level is $25$. If the screw gauge has a zero error of $-0.004 \, cm$,the correct diameter of the ball is: (in $, cm$)

In an experiment with Vernier callipers of least count $0.1\,mm$,when two jaws are joined together,the zero of the Vernier scale lies to the right of the zero of the main scale and the $6^{th}$ division of the Vernier scale coincides with a main scale division. While measuring the diameter of a spherical bob,the zero of the Vernier scale lies between $3.2\,cm$ and $3.3\,cm$ marks,and the $4^{th}$ division of the Vernier scale coincides with a main scale division. The diameter of the bob is measured as $.......\,cm$.

Two full turns of the circular scale of a screw gauge cover a distance of $1 \ mm$ on its main scale. The total number of divisions on the circular scale is $50$. The screw gauge has a zero error of $-0.03 \ mm$. While measuring the diameter of a thin wire,a student notes the main scale reading of $3 \ mm$ and the $35^{\text{th}}$ division of the circular scale coincides with the reference line of the main scale. The diameter of the wire is: (in $mm$)

Answer the following:
$(a)$ You are given a thread and a metre scale. How will you estimate the diameter of the thread?
$(b)$ $A$ screw gauge has a pitch of $1.0\; mm$ and $200$ divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale?
$(c)$ The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of $100$ measurements of the diameter expected to yield a more reliable estimate than a set of $5$ measurements only?