A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of crosssectional area \('a'\) is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is \((a\,<\,<\,A)\)

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be :

A gas mixture consists of \(2\) moles of oxygen and 4 moles of neon at temperature \(T\). Neglecting all vibrational modes, the total internal energy of the system will be \(...........\,RT\) 

In the given arrangement of a doubly inclined plane two blocks of masses \(\mathrm{M}\) and \(\mathrm{m}\) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is \(0.25\) . The value of \(\mathrm{m}\), for which \(\mathrm{M}=10\) \(\mathrm{kg}\) will move down with an acceleration of \(2 \mathrm{~m} / \mathrm{s}^2\), is _______. (take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and \(\left.\tan 37^{\circ}=3 / 4\right)\)

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :-

A prism of angle \(75^{\circ}\) and refractive index \(\sqrt{3}\) is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ___________ .
\(\left(\sin 15^{\circ}=0.25\right.\) and \(\left.\sin 25^{\circ}=0.43\right)\)

A straight wire \(AB\) of mass \(40\,g\) and length \(50\,cm\) is suspended by a pair of flexible leads in uniform magnetic field of magnitude \(0.40\,T\) as shown in the figure. The magnitude of the current required in the wire to remove the tension in the supporting leads is ...........\(A\). (Take \(g=10\,ms ^{-2}\) ).

Two blocks with masses 100 g and 200 g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E . The energy stored in B , when spring constants \(k _{ A }, k _{ B }\) of A and \(B\), respectively satisfy the relation \(4 k _{ A }=3 k _{ B }\), is :

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)

In the figure, a very large plane sheet of positive charge is shown. \(P _{1}\) and \(P _{2}\) are two points at distance \(l\) and \(2 \,l\) from the charge distribution. If \(\sigma\) is the surface charge density, then the magnitude of electric fields \(E_{1}\) and \(E_{2}\) at \(P _{1}\) and \(P _{2}\) respectively are

Two identical bodies \(A\) and \(B\) of equal masses have initial velocities \(\vec{v_1} = 4\hat{i}\) m/s and \(\vec{v_2} = 4\hat{j}\) m/s respectively. The body \(A\) has acceleration \(\vec{a_1} = 6\hat{i} + 6\hat{j}\) m/s\(^2\) while the acceleration of the other body \(B\) is zero. The centre of mass of the two bodies moves in __________ path.