$A$ screw gauge has some zero error but its value is unknown. We have two identical rods. When the first rod is inserted in the screw gauge,the state of the instrument is shown by diagram $(I)$. When both the rods are inserted together in series,the state is shown by diagram $(II)$. What is the zero error of the instrument in $mm$? Given: $1 \, M.S.D. = 100 \, C.S.D. = 1 \, mm$.

Thickness of a pencil measured by a screw gauge (least count $0.001 \ cm$) comes out to be $0.802 \ cm$. The percentage error in the measurement is (in $\%$)

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

The pitch and the number of divisions on the circular scale for a given screw gauge are $0.5\,mm$ and $100$ respectively. When the screw gauge is fully tightened without any object,the zero of its circular scale lies $3$ divisions below the mean line. The readings of the main scale and the circular scale for a thin sheet are $5.5\,mm$ and $48$ respectively. The thickness of this sheet is: (in $,mm$)

The main scale of a vernier caliper reads in millimeters and its vernier scale is divided into $8$ divisions,which coincide with $5$ divisions of the main scale. When the two jaws of the instrument touch each other,the zero of the vernier scale coincides with the zero of the main scale. $A$ rod is placed between the two jaws. It is observed that the zero of the vernier scale lies just to the left of the $36^{th}$ division of the main scale and the fourth division of the vernier scale coincides with a main scale division. The measured value is .......... $cm$.

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?