\(2\, kg\) of a monoatomic gas is at a pressure of \(4\times10^4\, N/m^2\). The density of the gas is \(8\, kg/m^3\). What is the order of energy of the gas due to its thermal motion ?

The de Broglie wavelengths of a proton and an \(\alpha\) particle are \(\lambda\) and \(2 \lambda\) respectively. The ratio of the velocities of proton and \(\alpha\) particle will be _______.

If \(1000\) droplets of water of surface tension \(0.07\,N / m\). having same radius \(1\,mm\) each, combine to from a single drop. In the process the released surface energy is \(\left(\text { Take } \pi=\frac{22}{7}\right)\)

The electric flux is \(\phi=\alpha \sigma+\beta \lambda\) where \(\lambda\) and \(\sigma\) are linear and surface charge density, respectively. \(\left(\frac{\alpha}{\beta}\right)\) represents

The resistance of an electrical toaster has a temperature dependence given by \(R\left( T \right) = {R_0}\left[ {1 + \alpha \left( {T - {T_0}} \right)} \right]\) in its range of operation. At \({T_0} = 300\,K,R = 100\,\Omega \) and at \(T = 500\,K,\,R = 120\,\Omega \). The toaster is connected to a voltage source at \(200\, V\) and its temperature is raised at a constant rate from \(300\) to \(500\, K\) in \(30\, s\). The total work done in raising the temperature is

A person travelling on a straight line moves with a uniform velocity \(\mathrm{v}_1\) for a distance x and with a uniform velocity \(\mathrm{v}_2\) for the next \(\frac{3}{2} \mathrm{x}\) distance. The average velocity in this motion is \(\frac{50}{7} \mathrm{~m} / \mathrm{s}\). If \(\mathrm{v}_1\) is \(5 \mathrm{~m} / \mathrm{s}\) then \(\mathrm{v}_2=\) _________ \(\mathrm{m} / \mathrm{s}\).

The formula \(X = 5YZ^2\),  \(X\) and \(Z\) have dimensions of capacitance and magnetic field respectively. What are the dimensions of \(Y\) in \(SI\) units?

A small ball of mass \(M\) and density \(\rho\) is dropped in a viscous liquid of density \(\rho_0\). After some time, the ball falls with a constant velocity. What is the viscous force on the ball ?

Applying the principle of homogeneity of dimensions, determine which one is correct. where \(\mathrm{T}\) is time period, \(\mathrm{G}\) is gravitational constant, \(M\) is mass, \(r\) is radius of orbit.

Let \(\vec A\, = \,(\hat i\, + \,\hat j)\,\)  and \(\vec B\, = \,(2\hat i\, - \,\hat j)\,.\)  The magnitude of a coplanar vector \(\vec C\) such that  \(\vec A\cdot \vec C\, = \,\vec B\cdot \vec C\, = \vec A\cdot \vec B\) is given by