The time period of simple harmonic motion of mass \(\mathrm{M}\) in the given figure is \(\pi \sqrt{\frac{\alpha M}{5 K}}\), where the value of \(\alpha\) is _______.

A boy is rolling a \(0.5\, kg\) ball on the frictionless floor with the speed of \(20\, ms ^{-1}\). The ball gets deflected by an obstacle on the way. After deflection it moves with \(5 \%\) of its initial kinetic energy. What is the speed of the ball now ? (in \(ms ^{-1}\))

A body of mass \(m\) is taken from the surface of earth to a height equal to twice the radius of earth (\(R_e\)). The increase in potential energy will be _______.
(\(g\) is acceleration due to gravity at the surface of earth)

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

In a physical balance working on the principle of moments, when \(5\, mg\) weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

Take the mean distance of the moon and the sun from the earth to be \(0.4 \times 10^6\,km\) and \(150 \times 10^6\,km\) respectively. Their masses are \(8 \times 10^{22}\, kg\) and \(2 \times 10^{30}\, kg\) respectively. The radius of the earth is \(6400\, km\). Let \(\Delta {F_1}\) be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and \(\Delta {F_2}\) be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then, the number closest to \(\frac{{\Delta {F_1}}}{{\Delta {F_2}}}\) is

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure \(90\) kPa and temperature \(400\) K. Keeping the temperature of one vessel constant at \(400\) K the second vessel temperature is raised to \(500\) K. The final pressure in the vessels is _______ kPa.

The temperature of a metal strip having coefficient of linear expansion \(\alpha\) is increased from \(T_1\) to \(T_2\) resulting in increase of its length by \(\Delta L_1\). The temperature is further increased from \(T_2\) to \(T_3\) such that the increase in its length is \(\Delta L_2\). Given \(T_3+T_1=2T_2\) and \(T_2-T_1=\Delta T\), the value of \(\Delta L_2\) is ______.

An inductor of inductance \(10\) mH having resistance of \(100\,\Omega\) is connected to battery of E.M.F. \(1.0\) V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is \(2\) mA and \(4\) mA is _______.

Two narrow bores of diameter \(5.0\, {mm}\) and \(8.0\, {mm}\) are joined together to form a \(U-\)shaped tube open at both ends. If this \({U}\)-tube contains water, what is the difference in the level of two limbs of the tube. [Take surface tension of water \({T}=7.3 \times 10^{-2} \, {Nm}^{-1}\), angle of contact \(=0, {g}=10\, {ms}^{-2}\) and density of water \(\left.=1.0 \times 10^{3} \,{kg} \,{m}^{-3}\right]\) (in \(mm\))

A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance \(2.5 \mu \mathrm{~F}\). The dielectric constant of the medium between the capacitor plates is 1 . It produces an instantaneous displacement current of 0.25 mA in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be _____ \(\mathrm{Vs}^{-1}\).

A hydrogen atom in its ground state absorbs \(10.2\ eV\) of energy. The orbital angular momentum is increased by (Given Planck constant \(h = 6.6 \times {10^{ - 34}}J -  sec\))

A student measured the length of a rod and wrote it as \(3.50\;cm\). Which instrument did he use to measure it?