A particle is moving in a circular path of radius $$a$$ under the action of an attractive potential U = - {k \over {2{r^2}}} Its total energy is:

Two particles of the same mass m are moving in circular orbits because of force, given by F\left( r \right) = {{ - 16} \over r} - {r^3} The first particle is at a distance r = 1, and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to :

A body of mass m starts moving from rest along x-axis so that its velocity varies as $$\upsilon =a\sqrt{s}$$ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

In an a.c. circuit, the instantaneous e.m.f. and current are given by e = 100 sin 30 t i = 20 sin \left( {30t - {\pi \over 4}} \right) In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively

For an RLC circuit driven with voltage of amplitude vm and frequency $${\omega }_0$$ = {1 \over {\sqrt {LC} }} the current exhibits resonance. The quality factor, Q is given by :

An ideal capacitor of capacitance $$0.2\mu F$$ is charged to a potential difference of $$10$$ $$V.$$ The charging battery is then disconnected. The capacitor is then connected to an ideal inductor of self inductance $$0.5$$ $$mH.$$ The current at a time when the potential difference across the capacitor is $$5$$ $$V,$$ is :

A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns, giving the output power at 230 V. If the current in the primary of the transformer is 5 A, and its efficiency is 90%, the output current would be ;

A carrier wave of peak voltage 14 V is used for transmitting a message signal. The peak voltage of modulating signal given to achieve a modulation index of 80% will be :

The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance 49 $$\mu$$H and a capacitance of 2.5 nF. It is modulated by an audio signal of $$12$$ $$kHz.$$ The frequency range occupied by the side bands is :

The number of amplitude modulted broadcast stations that can be accomodated in a $$300$$ $$kHz$$ band width for the highest modulating frequency $$15$$ $$kHz$$ will be :

A telephonic communication service is working at carrier frequency of 10 GHz. Only 10% of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of 5 kHz ?

A solution containing active cobalt $${}_{27}^{60}Co$$ having activity of $$0.8$$ $$\mu Ci$$ and decay constant $$\lambda$$ is injected in an animal's body. If $$1c{m}^3$$ of blood is drawn from the animal's body after $$10$$ hrs of injection, the activity found was $$300$$ decays per minute What is the volume of blood that is flowing in the body ? $$(Ci=3.7\times {10}^{10}$$ decays per second and at $$t=10$$ hrs $$e^{-\lambda t}=0.84)$$

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of 8 : 27. The ratio of the radii of the nuclei (assumed to be spherical) is :

Muon ($$\mu$$$$-$$) is a negatively charged (|q| = |e|) particle with a mass m$$\mu$$ = 200 me, where me is the mass of the electron and e is the electronic charge. If $$\mu$$$$-$$ is bond to a proton to form a hydrogen like atom, identify the correct statements. (A) Radis of the muonic orbit is 200 times smaller than that of the electron. (B) The speed of the $$\mu$$$$-$$ in the nth orbit is {1 \over {200}} times that of the electron in the nth orbit. (C) The ionization energy of muonic atom is 200 timesmore than of an hydroen atom. (D) The momentum of the muon in the nth orbit is 200 times more than that of the electron.

At some instant, a radioactive sample S1 having an activity 5 $$\mu$$Ci has twice the number of nuclei as another sample S2 which has an activity of 10 $$\mu$$Ci. The half lives of S1 and S2 are :

If the series limit frequency of the Lyman series is $${\nu }_L$$, then the series limit frequency of the Pfund series is:

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is pd; while for its similar collision with carbon nucleus at rest, fractional loss of energy is pc. The values of pd and pc are respectively :

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let $${\lambda }_n$$, $${\lambda }_g$$ be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let $${\Lambda }_n$$ be the wavelength of the emitted photon in the transition from the nth state to the ground state. For large n, (A, B are constants)

The energy required to remove the electron from a singly ionized Helium atom is $$2.2$$ times the energies required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is :

A Carnot's engine works as a refrigerator between $$250$$ K and $$300$$ K. It receives $$500$$ cal heat from the reservoir at the lower temperature. The amount of work done in each cycle to operate the refrigerator is :