A cell, shunted by a \(8 \; \Omega\) resistance, is balanced across a potentiometer wire of length \(3 \;  m\). The balancing length is \(2 \; m\) when the cell is shunted by \(4 \; \Omega\) resistance. The value of internal resistance of the cell will be \(\dots \; \Omega .\)

An object of mass 8 kg is hanging from one end of a uniform rod \(C D\) of mass 2 kg and length 1 m pivoted at its end \(C\) on a vertical wall as shown in figure. It is supported by a cable \(A B\) such that the system is in equilibrium. The tension in the cable is:
(Take \(g=10 m s ^{-2}\) )

Two electric bulbs, rated at \((25\, W, 220\, V)\) and \((100\, W, 220\, V)\), are connected in series acroos a \(220\, V\) voltage source. If the \(25\, W\) and \(100\, W\) bulbs draw powers \(P_1\) and \(P_2\) respectively, then

An \(LCR\) series circuit of capacitance \(62.5\,nF\) and resistance of \(50\,\Omega\). is connected to an \(A.C.\) source of frequency \(2.0\,kHz\). For maximum value of amplitude of current in circuit, the value of inductance is \(..........mH\). (take \(\left.\pi^2=10\right)\)

The voltage and the current between \(A\) and \(B\) points shown in the circuit are _____.

The initial pressure and volume of an ideal gas are \(P_0\) and \(V_0\). The final pressure of the gas when the gas is suddenly compressed to volume \(\frac{ V _0}{4}\) will be (Given \(\gamma=\) ratio of specific heats at constant pressure and at constant volume)

A square frame of side \(10\, cm\) and a long straight wire carrying current \(1\, A\) are in the plate of the paper. Starting from close to the wire, the frame moves towards the right with a constant speed of \(10\, ms^{-1}\) (see figure). The \(e.m. f\) induced at the time the left arm of the frame is at \(x\, = 10\, cm\) from the wire is .....\(\mu V\)

The wavelength of the radiation emitted is \(\lambda_0\) when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be \(\frac{20}{x} \lambda_0\). The value of \(x\) is \(........\)

Consider two blocks A and B of masses \(m_1=10 \mathrm{~kg}\) and \(m_2=5 \mathrm{~kg}\) that are placed on a frictionless table. The block A moves with a constant speed \(v=3 \mathrm{~m} / \mathrm{s}\) towards the block B kept at rest. A spring with spring constant \(\mathrm{k}=3000 \mathrm{~N} / \mathrm{m}\) is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is _______. (Neglect the mass of the spring)

Let \(\gamma_1\) be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and \(\gamma_2\) be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio, \(\frac{\gamma_1}{\gamma_2}\) is

Two sitar strings, \(A\)  and \(B,\)  playing the note \('Dha'\)  are slightly out of tune and produce beats and frequency  \(5\,Hz.\)  The tension of the string \(B\) is slightly increased and the beat frequency is found to decrease by  \(3\,Hz\) . If the frequency of \(A\) is \(425\,Hz,\)  the original frequency of \(B\) is  ... \(Hz\)

An automobile of mass \('m'\) accelerates starting from origin and initially at rest, while the engine supplies constant 10.Power \(P\). The position is given as a function of time by:

A thin lens made of glass (refractive index \(= 1.5\)) of focal length \(f=16\; \mathrm{cm}\) is immersed in a liquid of refractive index \(1.42 .\) If its focal length in liquid is \(f_{l},\) then the ratio \(f_{l} /f\) is closest to the integer