Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure. If the mass of each…

A small bob $A$ of mass $m$ is attached to a massless rigid rod of length 1 m pivoted at point $P$ and kept at an angle of $60^{\circ}$ with vertical as shown in figure. At distance of 1 m below point $P$, an identical bob $B$ is kept at rest on a smooth horizontal surface that extends to a circular track of radius $R$ as shown in figure. If bob $B$ just manages to complete the circular path of radius $R$ upto a point $Q$ after being hit elastically by $\operatorname{bob} A$, then radius $R$ is $\_\_\_\_$ m. 

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are $100$ divisions in circular scale and pitch of screw gauge is $0.1$ mm. When diameter of a sphere is measured, the reading of main scale is $5$ mm and $50^{\text{th}}$ division of circular scale coincides with the reference line of main scale. The diameter of sphere is _______ mm.

Two masses $m$ and 2 m are connected by a light string going over a pulley (disc) of mass 30 m with radius $r=0.1 \mathrm{~m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2 m mass is released from rest and its speed when it has descended through a height of 3.6 m is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$. (Assume string does not slip and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$)

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at $A$ and $B$ are $6 \mathrm{~cm}^{2}$ and $3 \mathrm{~cm}^{2}$ respectively. The rate of flow will be $\_\_\_\_$ $\mathrm{cm}^{3} / \mathrm{s}$. (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$) 

A lift of mass $1600$ kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4 \times 10^8$ N/m$^2$ and its radius is $4$ mm, then maximum acceleration the lift can take is _______ m/s$^2$. (take $g = 10$ m/s$^2$ and $\pi = 3.14$)