In an experiment for estimating the value of focal length of converging mirror, image of an object placed at $$40 \mathrm{~cm}$$ from the pole of the mirror is formed at distance $$120 \mathrm{~cm}$$ from the pole of the mirror. These distances are measured with a modified scale in which there are 20 small divisions in $$1 \mathrm{~cm}$$. The value of error in measurement of focal length of the mirror is $$\frac{1}{\mathrm{~K}} \mathrm{~cm}$$. The value of $$\mathrm{K}$$ is __________.

The general displacement of a simple harmonic oscillator is $$x = A\sin \omega t$$. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when $$t = {T \over \beta }$$. The value of $$\beta$$ is ______________.

In Young's double slit experiment, two slits $$S_{1}$$ and $$S_{2}$$ are '$$d$$' distance apart and the separation from slits to screen is $$\mathrm{D}$$ (as shown in figure). Now if two transparent slabs of equal thickness $$0.1 \mathrm{~mm}$$ but refractive index $$1.51$$ and $$1.55$$ are introduced in the path of beam $$(\lambda=4000$$ $$\mathop A\limits^o $$) from $$\mathrm{S}_{1}$$ and $$\mathrm{S}_{2}$$ respectively. The central bright fringe spot will shift by ___________ number of fringes.

A thin uniform rod of length $$2 \mathrm{~m}$$, cross sectional area '$$A$$' and density '$$\mathrm{d}$$' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $$\omega$$. If value of $$\omega$$ in terms of its rotational kinetic energy $$E$$ is $$\sqrt{\frac{\alpha E}{A d}}$$ then value of $$\alpha$$ is ______________.