Capacitance of an isolated conducting sphere of radius \(R_{1}\) becomes \(n\) times when it is enclosed by a concentric conducting sphere of radius \(R_{2}\) connected to earth. The ratio of their radii \(\left(\frac{ R _{2}}{ R _{1}}\right)\) is:

A vertical closed cylinder is separated into two parts by a frictionless piston of mass \(m\) and of negligible thickness. The piston is free to move along the length of the cylinder .The length of the cylinder above the piston is \(l_1,\) and that below the piston is \(l_2,\) such that \(l_1 > l_2.\) Each part of the cylinder contains \(n\) moles of an ideal gas at equal temperature \(T.\) If the piston is stationary, its mass, \(m,\) will be given by: ( \(R\) is universal gas constant and \(g\) is the acceleration due to gravity)

Two masses \(m\) and \(\frac{m}{2}\) are connected at the two ends of a massless rigid rod of length \(l\). The rod is suspended by a thin wire of torsional constant \(k\) at the centre of mass of the rodmass system (see figure). Because of torsional constant \(k\), the restoring torque is \(\tau  = k\,\theta \) for angular displacement \(\theta \). If the rod is rotated by \(\theta _0\) and released, the tension in it when it passes through its mean position will be

A square loop \(\mathrm{PQRS}\) having \(10\) turns, area \(3.6 \times\) \(10^{-3} \mathrm{~m}^2\) and resistance \(100 \Omega\) is slowly and uniformly being pulled out of a uniform magnetic field of magnitude \(\mathrm{B}=0.5 \mathrm{~T}\) as shown. Work done in pulling the loop out of the field in \(1.0 \mathrm{~s}\) is _______ \(\times 10^{-6} \mathrm{~J}\).

The bob of a pendulum was released from a horizontal position. The length of the pendulum is \(10 \mathrm{~m}\). If it dissipates \(10 \%\) of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is : _______. [Use, \(\mathrm{g}: 10 \mathrm{~ms}^{-2}\) ]

A hanging mass \(M\) is connected to a four times bigger mass by using a string-pulley arrangement. as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by \(2\,Mg\) force. In this situation. tension in the string is \(\frac{x}{5}\) \(Mg\) for \(x =\) Neglect mass of the string and friction of the block (bigger mass) with ice slab. (Given \(g=\) acceleration due to gravity)

If force \(\vec{F}=3 \hat{i}+4 \hat{j}-2 \hat{k}\) acts on a particle having position vector \(2 \hat{i}+\hat{j}+2 \hat{k}\) then, the torque about the origin will be

Small water droplets of radius \(0.01 \mathrm{~mm}\) are formed in the upper atmosphere and falling with a terminal velocity of \(10 \mathrm{~cm} / \mathrm{s}\). Due to condensation, if \(8 \mathrm{such}\) droplets are coalesced and formed a larger drop, the new terminal velocity will be _______ \(\mathrm{cm} / \mathrm{s}\).

A cubical block of side \(30\,cm\)  is moving with  velocity \(2\,\,ms^{-1}\) on a smooth horizontal surface. The surface has a bump at a point \(O\) as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is

Wires \({W}_{1}\) and \({W}_{2}\) are made of same material having the breaking stress of \(1.25 \times 10^{9} \,{N} / {m}^{2}\) \({W}_{1}\) and \({W}_{2}\) have cross-sectional area of \(8 \times 10^{-7}\, {m}^{2}\) and \(4 \times 10^{-7}\, {m}^{2}\), respectively. Masses of \(20 \,{kg}\) and \(10\, {kg}\) hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is \(.....{kg}\) (Use \(\left.{g}=10\, {m} / {s}^{2}\right)\)