A cylinder of mass m and radius R rolls down on an inclined plane of inclination $$\theta$$. Calculate the linear acceleration of axis of cylinder.

A long solenoid of radius a and number of turns per unit length $$n$$ is enclosed by cylindrical shell of radius R, thickness $$d$$ $$(d < < R)$$ and length L. A variable current $$\mathrm{I}=\mathrm{I}_{0} \sin \omega t$$ flows through the coil. If the resistivity of the material of cylindrical shell is $$\mathrm{P}$$, find the induced current in the shell.

Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass $$m$$ hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

Highly energetic electrons are bombarded on a target of an element containing 30 neutrons. The ratio of radii of nucleus to that of Helium nucleus is $$(14)^{\frac{1}{3}}$$. Find

(A) Atomic number of the nucleus;

(B) the frequency of $$\mathrm{K}_{\alpha}$$ line of the X-ray produced.

$$\left(\mathrm{R}=1.1 \times 10^{7} \mathrm{~m}^{-1}\right.$$ and $$\left.c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$$