A string is wrapped around the rim of a wheel of moment of inertia \(0.40 \mathrm{kgm}^2\) and radius \(10 \mathrm{~cm}\). The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of \(40 \mathrm{~N}\). The angular velocity of the wheel after \(10\) \(\mathrm{s}\) is \(\mathrm{x} \mathrm{rad} / \mathrm{s}\), where \(\mathrm{x}\) is _______.

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

Two projectile thrown at \(30^{\circ}\) and \(45^{\circ}\) with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is

The magnitude of the average electric field normally present in the atmosphere just above the surface of the Earth is about \(150\, N/C\), directed inward towards the center of the Earth . This gives the total net surface charge carried by the Earth to be......\(kC\) [Given \({\varepsilon _0} = 8.85 \times {10^{ - 12}}\,{C^2}/N - {m^2},{R_E} = 6.37 \times {10^6}\,m\)]

A thermometer graduated according to a linear scale reads a value \(x_0\) when in contact with boiling water, and \(x_0/3\) when in contact with ice. What is the temperature of an object in \({}^oC\), if this thermometer in the contact with the object reads \(x_0/2\) ?

A series combination of resistor of resistance \(100\,\Omega\), inductor of inductance \(1\,H\) and capacitor of capacitance \(6.25\,\mu F\) is connected to an ac source. The quality factor of the circuit will be \(.............\).

The resistance per centimeter of a meter bridge wire is \(\mathrm{r}\), with \(\mathrm{X}\  \Omega\) resistance in left gap. Balancing length from left end is at \(40 \mathrm{~cm}\) with \(25\  \Omega\) resistance in right gap. Now the wire is replaced by another wire of \(2 \mathrm{r}\) resistance per centimeter. The new balancing length for same settings will be at _______.

Let the first term \(a\) and the common ratio \(r\) of a geometric progression be positive integers. If the sum of its squares of first three terms is \(33033\), then the sum of these three terms is equal to

This questions has Statement \(I\) and Statement \(II\) Of the four choices given after the Statements, choose the one that best describes into two Statements.
Statement\(-I\) : Higher the range, greater is the resistance of ameter.
Statement\(-II\) : To increase the range of ammeter, additional shunt needs to be used across it.

The volume of an ideal gas increases 8 times and temperature becomes (1/4) of initial temperature during a reversible change. If there is no exchange of heat in this process (\( \Delta Q=0 \)) then identify the gas from the following options:

Let \([t]\) denote the greatest integer less than or equal to \(t.\) Then, the value of the integral \(\int\limits_{0}^{1}\left[-8 x^{2}+6 x-1\right] d x\) is equal to

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason(R).
Assertion (A) : Magnetic monopoles do not exist.
Reason (R) : Magnetic field lines are continuous and form closed loops.
In the light of the above statements, choose the most appropriate answer from the options given below :

Motion of a particle in \(x - y\) plane is described by a set of following equations \(x=4 \sin \left(\frac{\pi}{2}-\omega t\right) m\) and \(y=4 \sin (\omega t) m\). The path of particle will be