The radius of fifth orbit of the \(L ^{++}\)is \(.........\times 10^{-12}\,m\). Take : radius of hydrogen atom \(=0.51 \mathring A\)

A pipe open at both ends has a fundamental frequency \(f\) in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now ..... \(f\)

A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into \(a\) square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:

The machine as shown has \(2\) rods of length \(1\, m\) connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor  in a slot As the roller goes back and forth, a \(2\, kg\) weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a

A body having specific charge \(8\,\mu {C} / {g}\) is resting on a frictionless plane at a distance \(10\, {cm}\) from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of \(100 \,{V} / {m}\) is applied horizontally toward the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be \(....\, S.\)

A particle is released on a vertical smooth semicircular track from point \(X\) so that \(OX\) makes angle \(\theta \) from the vertical ( see figure). The normal reaction of the track on the particle vanishes at point \(Y\) where \(OY\) makes angle \(\phi \) with the horizontal. Then

Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If \(\sqrt{8}\, R\) is the distance between the centres of a \(mng\) (of mass ' \(m\) ) and a sphere (mass ' \(M\) ) where both have equal radius \('R'.\)

The magnetic field in a plane electromagnetic wave is \(\mathrm{B}_y=\left(3.5 \times 10^{-7}\right) \sin \left(1.5 \times 10^3 \mathrm{x}+0.5\right.\) \(\left.\times 10^{11} \mathrm{t}\right) \mathrm{T}\). The corresponding electric field will be _______.

The position of an object having mass \(0.1\) kg as a function of time \(t\) is given as \(\vec{r} = \left(10t^2\hat{i} + 5t^3\hat{j}\right)\) m. At \(t = 1\) s, which of the following statements are correct?
A. The linear momentum \(\vec{p} = \left(2\hat{i} + 1.5\hat{j}\right)\) kg·m/s.
B. The force acting on the object \(\vec{F} = \left(2\hat{i} + 3\hat{j}\right)\) N.
C. The angular momentum of the object about its origin \(\vec{L} = 15\hat{k}\) J·s.
D. The torque acting on the object about its origin \(\vec{\tau} = 20\hat{k}\) N·m.
Choose the correct answer from the options given below :

Two nuclei of mass number \(3\) combine with another nucleus of mass number \(4\) to yield a nucleus of mass number \(10\). If the binding energy per nucleon for the mass numbers \(3\), \(4\) and \(10\) are \(5.6\) MeV, \(7.4\) MeV and \(6.1\) MeV, respectively, then in the process, \(\Delta Mc^2 = \) _____ MeV.

Consider a finite insulated, uncharged conductor placed near a finite positively charged conductor. The uncharged body must have a potential

Two equal resistances when connected in series to a battery, consume electric power of \(60\,W.\) If these resistances are now connected in parallel combination to the same battery, the electric power consumed will be .............. \(W\)