In the arrangement shown in figure \(a _{1}, a _{2}, a _{3}\) and \(a _{4}\) are the accelerations of masses \(m _{1}, m _{2}, m _{3}\) and \(m _{4}\) respectively. Which of the following relation is true for this arrangement?

As shown in figure, when a spherical cavity (centred at \(\mathrm{O})\) of radius \(1\) is cut out of a uniform sphere of radius \(\mathrm{R} \text { (centred at } \mathrm{C}),\) the centre of mass of remaining (shaded) part of sphere is at \(G\), i.e, on the surface of the cavity. \(\mathrm{R}\) can be detemined by the equation

The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\)

One of the two identical conducing wires of length \(L\) is bent in the form of a circular loop and the other one into a circular coil of \(N\) identical turns. If the same current is passed in both, the ratio of the magnetic field at the central of the loop \((B_L)\) to that at the centre of the coil \((B_C),\); \(.\,\frac {B_L}{B_C}\) will be

The maximum speed of a boat in still water is \(27 \mathrm{~km} / \mathrm{h}\). Now this boat is moving downstream in a river flowing at \(9 \mathrm{~km} / \mathrm{h}\). A man in the boat throws a ball vertically upwards with speed of \(10 \mathrm{~m} / \mathrm{s}\). Range of the ball as observed by an observer at rest on the river bank, is _______ cm. (Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\))

A particle moving in the \(xy\) plane experiences a velocity dependent force \(\overrightarrow{ F }= k \left( v _{ y } \hat{ i }+ v _{ x } \hat{ j }\right),\) where \(v _{ x }\) and \(v _{ y }\) are the \(x\) and \(y\) components of its velocity \(\overrightarrow{ v } .\) If \(\overrightarrow{ a }\) be the acceleration of the particle, then which of the following statements is true for the particle\(?\)

In a Young's double slit experiment set up, the two slits are kept 0.4 mm apart and screen is placed at 1 m from slits. If a thin transparent sheet of thickness \(20 \mu m\) is introduced in front of one of the slits then centre bring fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by \(\frac{\alpha}{10}\), where \(\alpha\) is _________ .

An electric dipole has a fixed dipole moment \(\vec P\) which makes angle \(\theta \) with respect to \(x-\)axis. When subjected to an electric field \(\overrightarrow {{E_1}} \) \(=E\)\(\hat i\) it experiences a torque \(\overrightarrow {{T_1}} \) =\(\;\tau \hat k\) When subjected to another electric field \(\overrightarrow {{E_2}}  = \sqrt 3 {E_1}\hat j\) it experiences torque \(\overrightarrow {{T_2}} \) = \( - \overrightarrow {{T_1}} \) The angle \(\theta \;\) is.......\(^o\)

XPQY is a vertical smooth long loop having a total resistance R where PX is parallel to QY and separation between them is \(l\). A constant magnetic field B perpendicular to the plane of the loop exists in the entire space. A rod CD of length L (L > \(l\)) and mass m is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is __________ m/s.
(g = acceleration due to gravity)

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is \(200 \mathrm{~N} / \mathrm{m}\). The block is pushed such that the length of the spring becomes 1 m and then released. At distance \(\mathrm{x} \mathrm{m}(\mathrm{x} \lt 2)\) from the wall. the speed of the block will be :