What is the minimum energy required to launch a satellite of mass \(m\) from the surface of a planet of mass \(M\) and radius \(R\) in a circular orbit at an altitude of \(2R\)?

In a semiconductor, the number density of intrinsic charge carriers at \(27^{\circ} {C}\) is \(1.5 \times 10^{16} \,/ {m}^{3}\). If the semiconductor is doped with impurity atom, the hole density increase to \(4.5 \times 10^{22}\, / {m}^{3}\). The electron density in the doped semiconductor is \(.....\,\times 10^{9} / {m}^{3}\)

As shown below, bob \(A\) of a pendulum having massless string of length ' \(R\) ' is released from \(60^{\circ}\) to the vertical. It hits another bob \(B\) of half the mass that is at rest on a friction less table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.)

The amplitude of a particle executing \(SHM\) is \(3\,cm\). The displacement at which its kinetic energy will be \(25 \%\) more than the potential energy is: \(.............cm\).

A constant Power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time \('t'\) is proportional to

In the given circuit, the current in resistance \(\mathrm{R}_3 \) is _______.

For an ideal gas the instantaneous change in pressure \('{p} '\) with volume \('v'\) is given by the equation \(\frac{{dp}}{{dv}}=-{ap} .\) If \({p}={p}_{0}\) at \({v}=0\) is the given boundary condition, then the maximum temperature one mole of gas can attain is : (Here \({R}\) is the gas constant)

Starting at temperature \(300\; \mathrm{K},\) one mole of an ideal diatomic gas \((\gamma=1.4)\) is first compressed adiabatically from volume \(\mathrm{V}_{1}\) to \(\mathrm{V}_{2}=\frac{\mathrm{V}_{1}}{16} .\) It is then allowed to expand isobarically to volume \(2 \mathrm{V}_{2} \cdot\) If all the processes are the quasi-static then the final temperature of the gas (in \(\left. \mathrm{K}\right)\) is (to the nearest integer)

Electric charge is transferred to an irregular metallic disk as shown in figure. If \(\sigma_1, \sigma_2, \sigma_3\) and \(\sigma_4\) are charge densities at given points then, choose the correct answer from the options given below:

A metal block of mass \(m\) is suspended from a rigid support through a metal wire of diameter \(14\,mm\). The tensile stress developed in the wire under equilibrium state is \(7 \times 10^5\,Nm ^{-2}\). The value of mass \(m\) is \(......kg\). (Take, \(g =9.8\,ms ^{-2}\) and \(\left.\pi=\frac{22}{7}\right)\)

A resonance tube is old and has jagged end. It is still used in the laboratory to determine velocity of sound in air. A tuning fork of frequency \(512\,Hz\) produces first resonance when the tube is filled with water to a mark \(11\,cm\) below a reference mark, near the open end of the tube. The experiment is repeated with another fork of frequency \(256\,Hz\) which produces first resonance when water reaches a mark \(27\,cm\) below the reference mark. The velocity of sound in air, obtained in the experiment, is close to .... \(ms^{-1}\)