A block of mass \(M\) placed inside a box descends vertically with acceleration \('a'\). The block exerts a force equal to one-fourth of its weight on the floor of the box. The value of \('a'\) will be .............

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

A \(60\; HP\) electric motor lifts an elevator having a maximum total load capacity of \(2000\; \mathrm{kg}\). If the frictional force on the elevator is \(4000 \;\mathrm{N}\). the speed of the elevator at full load is close to ..............  \(\mathrm{m} / \mathrm{s}\) \(\left(1 \;\mathrm{HP}=746 \;\mathrm{W}, \mathrm{g}=10\; \mathrm{ms}^{-2}\right)\)

From the circuit given below, the capacitance between terminals \(A\) and \(B\) shown in the circuit is ______ \(\mu\)F.
(take \(C_1=C_2=C_3=1\) \(\mu\)F and \(C_4=2\) \(\mu\)F.)

As shown in figures, three identical polaroids \(P_1\), \(P _2\) and \(P _3\) are placed one after another. The pass axis of \(P_2\) and \(P_3\) are inclined at angle of \(60^{\circ}\) and \(90^{\circ}\) with respect to axis of \(P _1\). The source \(S\) has an intensity of \(256 \frac{ W }{ m ^2}\). The intensity of light at point \(O\) is \(...........\frac{ W }{ m ^2}\).

A pole is vertically submerged in swimming pool, such that it gives a length of shadow \(2.15 \,m\) within water when sunlight is incident at an angle of \(30^{\circ}\) with the surface of water. If swimming pool is filled to a height of \(1.5\,m\), then the height of the pole above the water surface in centimetres is \(\left(\eta_W=4 / 3\right)........\)

A bar magnet is released from rest along the axis of a very long vertical copper tube. After some time the magnet will

If \(\theta_{1}\) and \(\theta_{2}\) be respectively the smallest and the largest values of \(\theta\) in \((0,2 \pi)-\{\pi\}\) which satisfy the equation, \(\quad 2 \cot ^{2} \theta-\frac{5}{\sin \theta}+4=0,\) then \(\int\limits_{\theta_{1}}^{\theta_{2}} \cos ^{2} 3 \theta \mathrm{d} \theta \) is equal to

According to kinetic theory of gases, \(A\). The motion of the gas molecules freezes at \(0^{\circ} C\) \(B\). The mean free path of gas molecules decreases if the density of molecules is increased. \(C\). The mean free path of gas molecules increases if temperature is increased keeping pressure constant. \(D\). Average kinetic energy per molecule per degree of freedom is \(\frac{3}{2} k_{B} T\) (for monoatomic gases) Choose the most appropriate answer from the options given below

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\) Assertion \(A:\) The binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range \(30\) to \(170\). Reason \(R\) : Nuclear force is short ranged. In the light of the above statements, choose the correct answer from the options given below on

Let a curve \(y=f(x), x \in(0, \infty)\) pass through the points \(P\left(1, \frac{3}{2}\right)\) and \(Q\left(a, \frac{1}{2}\right)\). If the tangent at any point \(R(b, f(b))\) to the given curve cuts the \(y\)-axis at the point \(S(0, c)\) such that \(b c=3\), then \((P Q)^2\) is equal to \(.........\).

The dipole moment of a circular loop carrying a current \(I\), is \(m\) and the magnetic field at the centre of the loop is \(B_1\) . When the dipole moment is doubled by keeping the current constant, the magnetic field at the centre of the loop is \(B_2\) . The ratio \(\frac{{{B_1}}}{{{B_2}}}\) is

An effective power of a combination of \(5\) identical convex lenses which are kept in contact along the principal axis is \(25 \mathrm{D}\). Focal length of each of the convex lens is _______.