The vertical component of the earth's magnetic field is $$6 \times 10^{-5} \mathrm{~T}$$ at any place where the angle of dip is $$37^{\circ}$$. The earth's resultant magnetic field at that place will be $$\left(\right.$$Given $$\left.\tan 37^{\circ}=\frac{3}{4}\right)$$

The root mean square speed of smoke particles of mass $$5 \times 10^{-17} \mathrm{~kg}$$ in their Brownian motion in air at NTP is approximately. [Given $$\mathrm{k}=1.38 \times 10^{-23} \mathrm{JK}^{-1}$$]

Light enters from air into a given medium at an angle of $$45^{\circ}$$ with interface of the air-medium surface. After refraction, the light ray is deviated through an angle of $$15^{\circ}$$ from its original direction. The refractive index of the medium is:

A tube of length $$50 \mathrm{~cm}$$ is filled completely with an incompressible liquid of mass $$250 \mathrm{~g}$$ and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $$x \sqrt{F} \,\mathrm{rad} \,\mathrm{s}^{-1}$$. If $$\mathrm{F}$$ be the force exerted by the liquid at the other end then the value of $$x$$ will be __________.