A particle starts simple harmonic motion from the mean position. Its amplitude is \(a\) and total energy \(E\). At one instant its kinetic energy is \(3E/4.\) Its displacement at that instant is

Two resistors of \(200 \, \Omega\) and \(400 \, \Omega\) are connected in series with a battery of \(100\) V. A bulb rated at \(200\) V, \(100\) W is connected across the \(400 \, \Omega\) resistance. The potential drop across the bulb is _______ V.

From a tower of height \(H\), a particle is thrown vertically upwards with a speed \( u\). The time taken by the particle, to hit the ground, is \(n\) times that taken by it to reach the highest point of its path. The relation between \(H, u\) and \(n\) is:

A parallel plate capacitor is made of two circular plates separated by a distance \(5\ mm\) and with a dielectric of dielectric constant \(2.2\) between them. When the electric field in the dielectric is \(3 \times 10^4\) \( Vm^{-1}\) the charge density of the positive plate will be close to

A displacement current of \(4.0\) A can be set up in the space between two parallel plates of \(6\) \(\mu\)F capacitor. The rate of change of potential difference across the plates of the capacitor is nearly \(\alpha \times 10^6\) V/s. The value of \(\alpha\) is __________.

Two capacitors \(C_1\) and \(C_2\) are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are \(U_1\) and \(U_2\), respectively. Which of the given statements is true?

The internal energy of a monoatomic gas is 3 nRT . One mole of helium is kept in a cylinder having internal cross section area of \(17 cm^2\) and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J heat. If the temperature rises by \(4^{\circ} C\), then the piston will move ___________ cm.
(atmospheric pressure \(=10^5 Pa\) )

In the isothermal expansion of \(10\,g\) of gas from volume \(V\) to \(2V\) the work done by the gas is \(575\,J\). What is the root mean square speed of the molecules of the gas at that temperature ..... \(m/s\)?

The thermodynamic process, in which internal energy of the system remains constant is ...........

The acceleration due to gravity on the earth's surface at the poles is \(g\) and angular velocity of the earth about the axis passing through the pole is \(\omega .\) An object is weighed at the equator and at a height \(h\) above the poles by using a spring balance. If the weights are found to be same, then \(h\) is \(:( h << R ,\) where \(R\) is the radius of the earth)