A \(1 \mathrm{~kg}\) mass is suspended from the ceiling by a rope of length \(4 \mathrm{~m}\). A horizontal force ' \(F\) ' is applied at the mid point of the rope so that the rope makes an angle of \(45^{\circ}\) with respect to the vertical axis as shown in figure. The magnitude of \(F\) is _______.

A compound microscope is designed with two symmetric biconvex lenses. The objective lens is cut vertically, creating two identical plano-convex lenses. One of them is used in place of original objective lens. To retain same magnification keeping the object distance unchanged, the tube length has to be

In the circuit shown, current (in \(A\)) through \(50\; V\) and \(30\; V\) batteries are, respectively

Given below are two statements : \(Statement\) \((I)\) : Planck's constant and angular momentum have same dimensions. \(Statement\) \((II)\) : Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below :

Water from a pipe is coming at a rate of \(100\, litres\) per minute. If the radius of the pipe is \(5\, cm\), the Reynolds number for the flow is of the order of : (density of water \(= 1000\, kg/m^3\), coefficient of viscosity of water \(= 1\, mPa\, s\))

A small liquid drop of radius \(R\) is divided into \(27\) identical liquid drops. If the surface tension is \(\mathrm{T}\), then the work done in the process will be _______.

A thin metallic wire having cross sectional area of \(10^{-4} \mathrm{~m}^2\) is used to make a ring of radius \(30 \mathrm{~cm}\). A positive charge of \(2 \pi \mathrm{C}\) is uniformly distributed over the ring, while another positive charge of \(30\) \(\mathrm{pC}\) is kept at the centre of the ring. The tension in the ring is__________ \(\mathrm{N}\); provided that the ring does not get deformed (neglect the influence of gravity). (given, \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{SI}\) units)

A cube of metal is subjected to a hydrostatic pressure of \(4\;GPa.\) The percentage change in the length of the side of the cube is close to.......\(\%\)
(Given bulk modulus of metal, \(B =8 \times 10^{10}\, Pa\) )

A parallel - plate capacitor with plate area \(A\) has separation \(d\) between the plates. Two dielectric slabs of dielectric constant \({K}_{1}\) and \({K}_{2}\) of same area \(\frac A2\) and thickness \(\frac d2\) are inserted in the space between the plates. The capacitance of the capacitor will be given by :

A block of mass 1 kg , moving along x with speed \(\mathrm{v}_{\mathrm{i}}=10 \mathrm{~m} / \mathrm{s}\) enters a rough region ranging from \(\mathrm{x}=0.1 \mathrm{~m}\) to \(\mathrm{x}=1.9 \mathrm{~m}\). The retarding force acting on the block in this range is \(\mathrm{F}_{\mathrm{r}}=-\mathrm{kx} \mathrm{N}\), with \(\mathrm{k}=10 \mathrm{~N} / \mathrm{m}\). Then the final speed of the block as it crosses rough region is

A constant magnetic field of \(1 \,{T}\) is applied in the \({x}\,>\,0\) region. A metallic circular ring of radius \(1 \,{m}\) is moving with a constant velocity of \(1 \,{m} / {s}\) along the \(x\)-axis. At \(t=0\) \(s,\) the centre of \(O\) of the ring is at \({x}=-1 \,{m}\). What will be the value of the induced \(emf\) in the ring at \(t=1\, s\) ? (Assume the velocity of the ring does not change.) (In \({V}\))

Modern vacuum pumps can evacuate a vessel down to a pressure of \(4.0 \times {10^{ - 15}}\, atm\) at room temperature \((300\, K)\). Taking \(R = 8.0\, JK^{-1}\, mole^{-1}\) , \(1\, atm = 10^5\, Pa\) and \(N_ {Avogadro} = 6 \times 10^{23}\, mole^{-1}\) , the mean distance between molecules of gas in an evacuated vessel will be of the order of