A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in \(t_1\). If it is projected vertically downwards from the same point with the same speed, it reaches the ground in \(t_2\). Time required to reach the ground, if it is dropped from the top of the tower, is _______.

A particle of mass \(m\) moves in a circular orbit in a central potential field \(U ( r )= U _{0} r ^{4}\). If Bohr's quantization conditions are applied, radii of possible orbitals \(r _{ n }\) vary with \(n ^{1 / \alpha}\), where \(\alpha\) is ....... .

A block of mass \(m\) is placed on a surface having vertical cross section given by \(y=x^2 / 4\). If coefficient of friction is \(0.5\) , the maximum height above the ground at which block can be placed without slipping is _______.

This question has statement \(1\) and statement \(2\) . Of the four choices given after the statements, choose the one that best describes the two statements. 
Statement \(- 1\): A point particle of mass m moving with speed \(u\) collides with stationary point particle of mass \(M\). If the maximum energy loss possible is given as \(f\) \(\left( {\frac{1}{2}m{v^2}} \right)\) then \( f = \left( {\frac{m}{{M + m}}} \right)\)  Statement \(-2\): Maximum energy loss occurs when the particles get stuck together as a result of the collision.

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is \({W}\), then weight of the same object on that planet will be

A galvanometer of resistance \(100 \Omega\) when connected in series with \(400 \Omega\) measures a voltage of upto \(10 \mathrm{~V}\). The value of resistance required to convert the galvanometer into ammeter to read upto \(10 \mathrm{~A}\) is \(\mathrm{x} \times 10^{-2} \Omega\). The value of \(\mathrm{x}\) is _______.

Using a battery, a 100 pF capacitor is charged to 60 V and then the battery is removed. After that, a second uncharged capacitor is connected to the first capacitor in parallel. If the final voltage across the second capacitor is 20 V, its capacitance is : (in pF)

A conducting bar moves on two conducting rails as shown in the figure. A constant magnetic field B exists into the page. The bar starts to move from the vertex at time \(t=0\) with a constant velocity. If the induced EMF is \(\mathrm{E} \propto \mathrm{t}^{\mathrm{n}}\), then value of n is ________.

An aluminium wire is stretched to make its length, \(0.4 \,\%\) larger. Then percentage change in resistance is \(.....\,\%\)

A solid steel ball of diameter 3.6 mm acquired terminal velocity \(2.45 \times 10^{-2} \mathrm{~m} / \mathrm{s}\) while falling under gravity through an oil of density \(925 \mathrm{~kg} \mathrm{~m}^{-3}\). Take density of steel as \(7825 \mathrm{~kg} \mathrm{~m}^{-3}\) and g as 9.8 \(\mathrm{m} / \mathrm{s}^2\). The viscosity of the oil in SI unit is

An electron of mass ' m ' with an initial velocity \(\overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{i}\left(\mathrm{v}_0\gt0\right)\) enters an electric field \(\overrightarrow{\mathrm{E}}=-\mathrm{E}_{\mathrm{o}} \hat{\mathrm{k}}\). If the initial de Broglie wavelength is \(\lambda_0\), the value after time t would be _______.