Two slits are $$1 \mathrm{~mm}$$ apart and the screen is located $$1 \mathrm{~m}$$ away from the slits. A light of wavelength $$500 \mathrm{~nm}$$ is used. The width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern is __________ $$\times 10^{-4} \mathrm{~m}$$.

A potential divider circuit is connected with a dc source of $$20 \mathrm{~V}$$, a light emitting diode of glow in voltage $$1.8 \mathrm{~V}$$ and a zener diode of breakdown voltage of $$3.2 \mathrm{~V}$$. The length (PR) of the resistive wire is $$20 \mathrm{~cm}$$. The minimum length of PQ to just glow the LED is _________ $$\mathrm{cm}$$.

An object of mass $$0.2 \mathrm{~kg}$$ executes simple harmonic motion along $$x$$ axis with frequency of $$\left(\frac{25}{\pi}\right) \mathrm{Hz}$$. At the position $$x=0.04 \mathrm{~m}$$ the object has kinetic energy $$0.5 \mathrm{~J}$$ and potential energy $$0.4 \mathrm{~J}$$. The amplitude of oscillation is ________ $$\mathrm{cm}$$.

A circular table is rotating with an angular velocity of $$\omega \mathrm{~rad} / \mathrm{s}$$ about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of $$1 \mathrm{~m}$$ on the groove. All the surfaces are smooth. If the radius of the table is $$3 \mathrm{~m}$$, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $$x \sqrt{2} \omega \mathrm{~m} / \mathrm{s}$$, where the value of $$x$$ is _________.