The speed of a swimmer is \(4\,km\,h ^{-1}\) in still water. If the swimmer makes his strokes normal to the flow of river of width \(1\,km\), he reaches a point \(750 \,m\) down the stream on the opposite bank.The speed of the river water is \(.........km h ^{-1}\)

Two discs of moment of inertia \(I_1=4 \mathrm{~kg} \mathrm{~m}^2\) and \(\mathrm{I}_2=2 \mathrm{~kg} \mathrm{~m}^2\) about their central axes & normal to their planes, rotating with angular speeds \(10 \mathrm\ {rad} / \mathrm{s}\) & \(4  \mathrm\ {rad} / \mathrm{s}\) respectively are brought into contact face to face with their axe of rotation coincident. The loss in kinetic energy of the system in the process is _______ \(\mathrm{J}\).

A parallel plate capacitor is made of two square plates of side \(a\), separated by a distance \(d\,(d  < < a)\). The lower triangular portion is filled with a dielectric of dielectric constant \(K\), as shown in the figure. Capacitance of this capacitor is

A solid cylinder having radius \(R\) and length \(L\) is slipping on a rough horizontal plane. At time \(t=0\) the cylinder has a translational velocity \(v_0=49\) m/s, perpendicular to its axis and a rotational velocity \(v_0/4R\) about the centre. The time taken by the cylinder to start rolling is ________ seconds. (coefficient of kinetic friction \(\mu_K=0.25\) and \(g=9.8\) m/s\(^2\))

A transverse wave travels on a taut steel wire with a velocity of \({v}\) when tension in it is \(2.06 \times 10^{4} \;\mathrm{N} .\) When the tension is changed to \(T\). the velocity changed to \(\frac v2\). The value of \(\mathrm{T}\) is close to

In the reported figure, two bodies \(A\) and \(B\) of masses \(200\, {g}\) and \(800\, {g}\) are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be \(.....\,{rad} / {s}\) when \({k}=20 \,{N} / {m} .\)

A pendulume clock loses \(12\;s\) a day if the temperature is \(40^oC\) and gains \(4\;s\) a day if the temperature is \(20^oC\). The temperature at which the clock will show correct time, and the coeffecient of linear expansion \((\alpha)\) of the metal of the pendulum shaft are respectively

This question has statement \(1\) and statement \(2\) . Of the four choices given after the statements, choose the one that best describes the two statements. 
Statement \(- 1\): A point particle of mass m moving with speed \(u\) collides with stationary point particle of mass \(M\). If the maximum energy loss possible is given as \(f\) \(\left( {\frac{1}{2}m{v^2}} \right)\) then \( f = \left( {\frac{m}{{M + m}}} \right)\)  Statement \(-2\): Maximum energy loss occurs when the particles get stuck together as a result of the collision.

A particular hydrogen - like ion emits the radiation of frequency \(3 \times 10^{15} \mathrm{~Hz}\) when it makes transition from \(n=2\) to \(n=1\). The frequency of radiation emitted in transition from \(n=3\) to \(n=1\) is \(\frac{x}{9} \times 10^{15} \mathrm{~Hz}\), when \(x=\) = _____.

A cylindrical vessel of cross-section \(A\) contains water to a height \(h\) . There is a hole in the bottom of radius \('a'\) . The time in which it will be emptied is

A beam of electrons of energy \(E\) scatters from a target having atomic spacing of \(1\, A\). The first maximum intensity occurs at \(\theta=60^{\circ} .\) Then \(E\) (in \(eV )\) is\(......\) (Planck constant \(h =6.64 \times 10^{-34}\, Js\) \(1 \,eV =1.6 \times 10^{-19}\,J\), electron mass \(\left. m =9.1 \times 10^{-31}\, kg \right)\)