In a single slit diffraction pattern, a light of wavelength 6000$$\mathop A\limits^o$$ is used. The distance between the first and third minima in the diffraction pattern is found to be $$3 \mathrm{~mm}$$ when the screen in placed $$50 \mathrm{~cm}$$ away from slits. The width of the slit is _________ $$\times 10^{-4} \mathrm{~m}$$.

A body of mass $$5 \mathrm{~kg}$$ moving with a uniform speed $$3 \sqrt{2} \mathrm{~ms}^{-1}$$ in $$X-Y$$ plane along the line $$y=x+4$$. The angular momentum of the particle about the origin will be _________ $$\mathrm{kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$$.

Hydrogen atom is bombarded with electrons accelerated through a potential difference of $$\mathrm{V}$$, which causes excitation of hydrogen atoms. If the experiment is being performed at $$\mathrm{T}=0 \mathrm{~K}$$, the minimum potential difference needed to observe any Balmer series lines in the emission spectra will be $$\frac{\alpha}{10} \mathrm{~V}$$, where $$\alpha=$$ __________.

A simple harmonic oscillator has an amplitude $$A$$ and time period $$6 \pi$$ second. Assuming the oscillation starts from its mean position, the time required by it to travel from $$x=$$ A to $$x=\frac{\sqrt{3}}{2}$$ A will be $$\frac{\pi}{x} \mathrm{~s}$$, where $$x=$$ _________.