A metal wire of length \(0.5 m\) and cross-sectional area \(10^{-4}\; m ^{2}\) has breaking stress \(5 \times 10^{8}\; Nm ^{-2}\). A block of \(10\; kg\) is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be \(ms ^{-1}\).

In a screw gauge, there are \(100\) divisions on the circular scale and the main scale moves by \(0.5\,mm\) on a complete rotation of the circular scale. The zero of circular scale lies \(6\) divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, \(4\) linear scale divisions are clearly visible while \(46^{\text {th }}\) division the circular scale coincide with the reference line. The diameter of the wire is \(...........\times 10^{-2}\,mm\)

Position of an ant ( \(\mathrm{S}\) in metres) moving in \(\mathrm{Y}-\mathrm{Z}\) plane is given by \(S=2 t^2 \hat{j}+5 \hat{k}\) (where \(t\) is in second). The magnitude and direction of velocity of the ant at \(t=1 \mathrm{~s}\) will be :

As shown in the figure, two parallel plate capacitors having equal plate area of \(200\,cm ^2\) are joined in such a way that \(a \neq b\). The equivalent capacitance of the combination is \(x \varepsilon_0 F\). The value of \(x\) is \(..........\).

An ac circuit has an inductor and a resistor of resistance \({R}\) in series, such that \({X}_{{L}}=3 {R}\). Now, a capacitor is added in series such that \({X}_{{C}}=2 {R}\). The ratio of new power factor with the old power factor of the circuit is \(\sqrt{5}: {x}\). The value of \({x}\) is ...... .

A thin uniform rod \(( X )\) of mass M and length L is pivoted at a height \(\left(\frac{ L }{3}\right)\) as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is _________ . (g = gravitational acceleration)

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point \(P\) is \(\frac{x \sigma}{\epsilon_0}\). The value of \(x\) is _______. (all quantities are measured in \(SI\) units).

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass \(M\). The piston and the cylinder have equal cross sectional area \(A\). When the piston is in equilibrium, the volume of the gas is \(V_0\) and its pressure is \(P_ 0\). The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

When a light of a given wavelength falls on a metallic surface the stopping potential for photoelectrons is 3.2 V . If a second light having wavelength twice of first light is used, the stopping potential drops to 0.7 V . The wavelength of first light is _________ m .
\(\left( h =6.63 \times 10^{-34} J . s , e =1.6 \times 10^{-19} C , c =3 \times 10^8 m / s \right)\)

The magnetic field of a plane electromagnetic wave is given by \(\overrightarrow{ B }=2 \times 10^{-8} \sin \left(0.5 \times 10^{3} x +1.5 \times 10^{11} t \right) \hat{ j } T\) The amplitude of the electric field would be.