A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is

Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.Question:
The ratio \(\frac{x_1}{x_2}\) is

A ball of mass \((m) 0.5 \mathrm{~kg}\) is attached to the end of a string having length \((L) 0.5 \mathrm{~m}\). The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is \(324 \mathrm{~N}\). The maximum possible value of angular velocity of ball (in rad/s) is

A pendulum consists of a bob of mass \(m=0.1 \mathrm{~kg}\) and a massless inextensible string of length \(L=1.0 \mathrm{~m}\). It is suspended from a fixed point at height \(H=0.9 \mathrm{~m}\) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \(P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \(J \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{s}\). The kinetic energy of the pendulum just after the liftoff is \(K\) Joules.
The value of $K$ is ____.

Statement 1 The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
and Statement 2 In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, \(A\) is the point of contact. \(B\) is the centre of the sphere and \(C\) is its topmost point. Then,

A light disc made of aluminum (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

Match the integrals in Column I with the values in Column II.

An organic compound \(\mathbf{P}\) having molecular formula \(\mathrm{C}_6 \mathrm{H}_6 \mathrm{O}_3\) gives ferric chloride test and does not have intramolecular hydrogen bond. The compound \(\mathbf{P}\) reacts with 3 equivalents of \(\mathrm{NH}_2 \mathrm{OH}\) to produce oxime \(\mathbf{Q}\). Treatment of \(\mathbf{P}\) with excess methyl iodide in the presence of \(\mathrm{KOH}\) produces compound \(\mathbf{R}\) as the major product. Reaction of \(\mathbf{R}\) with excess iso-butylmagnesium bromide followed by treatment with \(\mathrm{H}_3 \mathrm{O}^{+}\)gives compound \(\mathrm{S}\) as the major product.
The total number of methyl \(\left(-\mathrm{CH}_3\right)\) group(s) in compound \(\mathbf{S}\) is ________. (consider maximum value if there are two values)

The correct statement $\left(\mathrm{s}\right)$ related to the metal extraction processes is(are)

A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{}\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{}\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{}\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____
[Speed of the sound $=330\mathrm{}\mathrm{m}/\mathrm{s}]$

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?