The displacement of a damped harmonic oscillator is given by \(x\left( t \right) = {e^{ - 0.1\,t}}\,\cos \left( {10\pi t + \varphi } \right)\)  The time taken for its amplitude of vibration to drop to half of its initial value is close to .... \(s\)

If \(\mathrm{n}\) is the number density and \(\mathrm{d}\) is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by _______.

The momentum of an electron revolving in \(n ^{\text {th }}\) orbit is given by: (Symbols have their usual meanings)

Sun light falls normally on a surface of area \(36\,cm ^{2}\) and exerts an average force of \(7.2 \times 10^{-9}\,N\) within a time period of \(20\) minutes. Considering a case of complete absorption, the energy flux of incident light is.

According to kinetic theory of gases, \(A\). The motion of the gas molecules freezes at \(0^{\circ} C\) \(B\). The mean free path of gas molecules decreases if the density of molecules is increased. \(C\). The mean free path of gas molecules increases if temperature is increased keeping pressure constant. \(D\). Average kinetic energy per molecule per degree of freedom is \(\frac{3}{2} k_{B} T\) (for monoatomic gases) Choose the most appropriate answer from the options given below

A nucleus disintegrates into two smaller parts, which have their velocities in the ratio \(3: 2\). The ratio of their nuclear sizes will be \(\left(\frac{x}{3}\right)^{\frac{1}{3}}\). The value of ' \(x\) ' is :

As shown in the figure, a particle is moving with constant speed \(\pi\,m / s\). Considering its motion from \(A\) to \(B\), the magnitude of the average velocity is:

An \(LCR\) circuit as shown in the figure is connected to a voltage source \(V_{ac}\) whose frequency can be varied. The frequency, at which the voltage across the resistor is maximum, is......\(Hz\)

The energy \(E\) and momentum \(p\) of a moving body of mass \(m\) are related by some equation. Given that c represents the speed of light, identify the correct equation.

The ratio of the power of a light source \(S_1\) to that the light source \(S_2\) is \(2 . S_1\) is emitting \(2 \times 10^{15}\) photons per second at 600 nm . If the wavelength of the source \(S_2\) is 300 nm , then the number of photons per second emitted by \(S_2\) is _______ \(\times 10^{14}\).

If \(x\) and \(y\) coordinates of a projectile as a function of time \((t)\) are given as \(24t\) and \(43.6t-4.9t^2\), respectively, then the angle (in degrees) made by the projectile with horizontal when \(t=2\) s is ______.

The radius ( \(\mathrm{r})\), length \((l)\) and resistance \((\mathrm{R})\) of a metal wire was measured in the laboratory as \(\mathrm{r}=(0.35 \pm 0.05) \mathrm{cm}\) \(\mathrm{R}=(100 \pm 10) \mathrm{ohm}\) \(l=(15 \pm 0.2) \mathrm{cm}\) The percentage error in resistivity of the material of the wire is _______.