A particle of mass \(m\) with an initial velocity \(u\hat i\) collides perfectly elastically with a mass \(3m\) at rest. It moves with a velocity \(v\hat j\) after collision, then, \(v\) is given by

A body of mass \(\mathrm{m}=10\; \mathrm{kg}\) is attached to one end of a wire of length \(0.3\; \mathrm{m} .\) The maximum angular speed (in \(rad \;s^{-1}\) ) with which it can be rotated about its other end in space station is (Breaking stress of wire \(=4.8 \times 10^{7} \;\mathrm{Nm}^{-2}\) and area of cross-section of the wire \(=10^{-2}\; \mathrm{cm}^{2}\) ) is

A small square loop of wire of side \(\ell\) is placed inside a large square loop of wire of side \(L\) \(\left(\mathrm{L}=\ell^2\right)\). The loops are coplanar and therr centers coinside. The value of the mutual inductance of the system is \(\sqrt{\mathrm{x}} \times 10^{-7} \mathrm{H}\), where X = _______.

If \(50\) Vernier divisions are equal to \(49\) main scale divisions of a travelling microscope and one smallest reading of main scale is \(0.5 \mathrm{~mm}\), the Vernier constant of travelling microscope is _______.

Two point dipoles of dipole moment \({\vec P_1}\) and \({\vec P_2}\) are at a distance \(x\) from each other and  \({\vec P_1}\) || \({\vec P_2}\). The force between the dipoles is

Expression for an electric field is given by \(\vec{E}=4000 x^2 \hat{i} \frac{V}{m}\). The electric flux through the cube of side \(20\,cm\) when placed in electric field (as shown in the figure) is \(.........V cm\).

If average depth of an ocean is \(4000 \mathrm{~m}\) and the bulk modulus of water is \(2 \times 10^9 \mathrm{Nm}^{-2}\), then fractional compression \(\frac{\Delta V}{V}\) of water at the bottom of ocean is \(\alpha \times 10^{-2}\). The value of \(\alpha\) is ___________(Given, \(\mathrm{g}=10 \mathrm{~ms}^{-2}, \rho=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) )

An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section \(8.0\times10^{-3}\, m^2\) . Initially the gas is at \(300\, K\) and occupies a volume of \(2.4\times10^{-3}\, m^3\) and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by \(0.1\, m\). The force constant of the spring is \(8000\, N/m\) and the atmospheric pressure is \(1.0\times10^5\, N/m^2\) . The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater . The heat capacity of the heater coil is also negligible)

Identify the logic operation carried out.

A diatomic gas \((\gamma=1.4)\) does \(200 \mathrm{~J}\) of work when it is expanded isobarically. The heat given to the gas in the process is _______.

\(A, B, C\) and \(D\) are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation \(AD = C\, ln\, (BD)\) holds true. Then which of the combination is not a meaningful quantity ?

The number of distinct real roots of \(x ^{4}-4 x +1=0\) is

An organ pipe \(40\,cm\) long is open at both ends. The speed of sound in air is \(360\,ms ^{-1}\). The frequency of the second harmonic is \(...........\,Hz\).