A pendulum of length \(2 \; m\) consists of a wooden bob of mass \(50 \; g\). A bullet of mass \(75 \; g\) is fired towards the stationary bob with a speed \(v\). The bullet emerges out of the bob with a speed \(\frac{v}{3}\) and the bob just completes the vertical circle. The value of \(v\) is \(\dots \; ms ^{-1}\). (if \(g =10 \; m / s ^{2}\) )

A light wave is incident normally on a glass slab of refractive index \(1.5\). If \(4\%\) of light gets reflected and the amplitude of the electric field of the incident light is \(30\, V/m\), then the amplitude of the electric field for the wave propagating in the glass medium will be.......\( V/m\)

The moment of inertia of a uniform cylinder of length \(l\) and radius \(R\) about its perpendicular bisector is \(I\). What is the ratio \(l/R\) such that the moment of inertia is minimum?

Two light beams of intensities \(4\,I\) and \(9\,I\) interfere on a screen. The phase difference between these beams on the screen at point \(A\) is \(zero\) and at point \(B\) is \(\pi\). The difference of resultant intensities, at the point \(A\) and \(B\), will be \(....I\).

A parallel beam of light is allowed to fall on a transparent spherical globe of diameter \(30 \,cm\) and refractive index \(1.5\). The distance from the centre of the globe at which the beam of light can converge is________ \(mm\)

In an experiment to measure the focal length ( \(f\) ) of a convex lens, the magnitude of object distance \((x)\) and the image distance \((y)\) are measured with reference to the focal point of the lens. The \(y-x\) plot is shown in figure. The focal length of the lens is _______ \(\mathrm{cm}\).

A solid sphere of mass \(500\,g\) and radius \(5\,cm\) is rotated about one of its diameter with angular speed of \(10\,rad \, s ^{-1}\). If the moment of inertia of the sphere about its tangent is \(x \times 10^{-2}\) times its angular momentum about the diameter. Then the value of \(x\) will be ..............

Which of the following statement is not true about stopping potential \(\left(\mathrm{V}_0\right)\)?

The gap between the plates of a parallel plate capacitor of area \(A\) and distance between plates \(d\), is filled with a dielectric whose permittivity varies linearly from \({ \varepsilon _1}\) at one plate to \({ \varepsilon _2}\) at the other. The capacitance of capacitor is

A solid sphere of mass \(‘M’\) and radius \(‘a’\) is surrounded by a uniform concentric spherical shell of thickness \(2a\) and mass \(2M.\) The gravitational field at distance \(‘3a’\) from the centre will be

Suppose two planets (spherical in shape) of radii \({R}\) and \(2 {R}\), but mass \({M}\) and \(9\, {M}\) respectively have a centre to centre separation \(8\, {R}\) as shown in the figure. A satellite of mass \('{m}'\) is projected from the surface of the planet of mass \('M'\) directly towards the centre of the second planet. The minimum speed \('v'\) required for the satellite to reach the surface of the second planets is \(\sqrt{\frac{a}{7} \frac{G M}{R}}\) then the value of \('a'\) is \(....\) [Given: The two planets are fixed in their position]

Four identical hollow cylindrical columns of mild steel support a big structure of mass \(50 \times 10^{3} {kg}\), The inner and outer radii of each column are \(50\; {cm}\) and \(100 \;{cm}\) respectively. Assuming uniform local distribution, calculate the compression strain of each column. [Use \(\left.{Y}=2.0 \times 10^{11} \;{Pa}, {g}=9.8\; {m} / {s}^{2}\right]\)

A person travelling on a straight line moves with a uniform velocity \(\mathrm{v}_1\) for a distance x and with a uniform velocity \(\mathrm{v}_2\) for the next \(\frac{3}{2} \mathrm{x}\) distance. The average velocity in this motion is \(\frac{50}{7} \mathrm{~m} / \mathrm{s}\). If \(\mathrm{v}_1\) is \(5 \mathrm{~m} / \mathrm{s}\) then \(\mathrm{v}_2=\) _________ \(\mathrm{m} / \mathrm{s}\).