A uniform rope of length '$$L$$' and mass '$$m_1$$' hangs vertically from a rigid support. A block of mass '$$m_2$$' is attached to the free end of the rope. A transverse wave of wavelength '$$\lambda_1$$' is produced at the lower end of the rope. The wavelength of the wave when it reaches the top of the rope is '$$\lambda_2$$'. The ratio $$\frac{\lambda_1}{\lambda_2}$$ is

In a vessel, the ideal gas is at a pressure $$\mathrm{P}$$. If the mass of all the molecules is halved and their speed is doubled, then resultant pressure of the gas will be

Two concentric circular coils having radii $$r_1$$ and $$r_2\left(r_2 << r_1\right)$$ are placed co-axially with centres coinciding. The mutual induction of the arrangement is (Both coils have single turn, $$\mu_0=$$ permeability of free space)

A system consists of three particles each of mass '$$m_1$$' placed at the corners of an equilateral triangle of side '$$\frac{\mathrm{L}}{3}$$', A particle of mass '$$\mathrm{m}_2$$' is placed at the mid point of any one side of the triangle. Due to the system of particles, the force acting on $$\mathrm{m}_2$$ is