A body of mass \(1 kg\) begins to move under the action of a time dependent force \(\vec{F}=\left(t \hat{i}+3 t^2 \hat{ j }\right) N\). where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along \(x\) and \(y\) axis. The 10.Power developed by above force, at the time \(t=2 s\). will be \(..............W\).

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights \(h_{sph}\) and \(h_{cyl}\) on the incline. The radio \(\frac{{{h_{sph}}}}{{{h_{cyl}}}}\) is given by

When Uranium is bombarded with neutrons, it undergoes fission. The fission reaction can be written as  \({}_{92}{U^{235}} + {}_0{n^1} \to {}_{56}B{a^{141}} + {}_{36}K{r^{92}} + 3x + Q{\rm{( energy)}}\) where three particles named \(x\) are produced and energy \(Q\) is released. What is the name of the particle \(x\) ?

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by \(y\left( {x,t} \right) = 0.5\sin\, \left( {\frac{{5\pi }}{4}x} \right)\,\cos\, \left( {200\,\pi t} \right)\). What is the speed of the travelling wave moving in the positive \(x\) direction .... \(m/s\) ? (\(x\) and \(t\) are in meter and second, respectively.)

At \(300\,K\), the rms speed of oxygen molecules is \(\sqrt{\frac{\alpha+5}{\alpha}}\) times to that of its average speed in the gas. Then, the value of \(\alpha\) will be (used \(\pi=\frac{22}{7}\) )

A thin metallic wire having cross sectional area of \(10^{-4} \mathrm{~m}^2\) is used to make a ring of radius \(30 \mathrm{~cm}\). A positive charge of \(2 \pi \mathrm{C}\) is uniformly distributed over the ring, while another positive charge of \(30\) \(\mathrm{pC}\) is kept at the centre of the ring. The tension in the ring is__________ \(\mathrm{N}\); provided that the ring does not get deformed (neglect the influence of gravity). (given, \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{SI}\) units)

\(N\,moles\) of a diatomic gas in a cylinder are at a temperature \(T\). Heat is supplied to the cylinder such that the temperature remains constant but \(n\,moles\) of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas ?

In a closed organ pipe, the frequency of fundamental note is \(30 \mathrm{~Hz}\). A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to \(110 \mathrm{~Hz}\). If the organ pipe has a cross-sectional area of \(2 \mathrm{~cm}^2\), the amount of water poured in the organ tube is _______ \(g.\) (Take speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\) )

A light rope is wound around a hollow cylinder of mass \(5\,kg\) and radius \(70\,cm\). The rope is pulled with a force of \(52.5\,N\). The angular acceleration of the cylinder will be.....\( {rad \,s}{ }^{-2}\).

The correct match between the entries in column \(I\) and column \(II\) are              \(I\)                         \((II)\) Radiation                     Wavelength \((a)\) Microwave          \((i)\) \(100 m\) \((b)\) Gamma rays      \((ii)\) \(10^{-15} m\) \((c)\) \(A.M.\) radio waves        \((iii)\) \(10^{-10} m\) \((d)\) \(X-\)rays             \((iv)\) \(10^{-3} m\)

A radioactive substance decays to \(\left(\frac{1}{16}\right)^{t h}\) of its initial activity in \(80\, days.\) The half life of the radioactive substance expressed in days is ... .

An infinite wire has a circular bend of radius a, and carrying a current I as shown in figure. The magnitude of magnetic field at the origin \(O\) of the arc is given by :