An expression of energy density is given by \(u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)\), where \(\alpha, \beta\) are constants, \(x\) is displacement, \(k\) is Boltzmann constant and \(t\) is the temperature. The dimensions of \(\beta\) will be.

A thin disc of radius \(b = 2a\)  has a concentric hole of radius \('a'\) in it (see figure). It carries uniform surface charge \('\sigma '\) on it. If the electric field on its axis at height \('h'\) \((h << a)\) from its centre is given as \('Ch'\) then value of \('C'\) is

A car moving with a speed of \(54\) km/h takes a turn of radius \(20\) m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take \(g=10\) m/s\(^2\))

A moving proton and electron have the same deBroglie wavelength. If \({K}\) and \({P}\) denote the \({KE}\). and momentum respectively. Then choose the correct option :

In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked \(A. B, C. D\) and \(E.\) The transitions \(A , B\) and \(C\) respectively represent

A uniform disc with mass \(M=4\,kg\) and radius \(R=\) \(10\,cm\) is mounted on a fixed horizontal axle as shown in figure. A block with mass \(m =2\,kg\) hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is_______ \(N\) \(\left(\right.\) Take \(\left.g =10\,ms ^{-2}\right)\)

An ideal gas at pressure \(P\) and temperature \(T\) is expanding such that \(PT^3 = \) constant. The coefficient of volume expansion of the gas is _______.

Given below are two statements: One is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A:\) For measuring the potential difference across a resistance of \(600\,\Omega\), the voltmeter with resistance \(1000\,\Omega\) will be preferred over voltmeter with resistance \(4000\,\Omega\). Reason \(R:\) Voltmeter with higher resistance will draw smaller current than voltmeter with lower resistance. In the light of the above statements, choose the most appropriate answer from the options given below.

A solid conducting sphere, having a charge \(Q\), is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be \(V\). If the shell is now given a charge of \(-4\, Q\), the new potential difference between the same two surface is......\(V\)

The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is _______.

Consider a uniform wire of mass \(M\) and length \(L\). It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :

A capacitor is discharging through a resistor \(R\). Consider in time \(t _{1}\), the energy stored in the capacitor reduces to half of its initial value and in time \(t_{2}\), the charge stored reduces to one eighth of its initial value. The ratio \(t_{1} / t_{2}\) will be ................