The flexural rigidity $$(EI)$$ of a cantilever beam is assumed to be constant over the length of the beam as shown in figure. If a load $$P$$ and bending moment $$PL/2$$ are applied at the free end of the beam then the value of the slope at the free end is

Consider a flywheel whose mass $$M$$ is distributed almost equally between a heavy, ring-like rim of radius $$R$$ and a concentric disk-like feature of radius $$R/2$$. Other parts of the flywheel, such as spokes, etc, have negligible mass. The best approximation for $$\alpha ,$$ if the moment of inertia of the flywheel about its axis of rotation is expressed as $$\,\alpha M{R^2},\,\,$$ is ________.

Maximum fluctuation of kinetic energy in an engine has been calculated to be $$2600$$ J. Assuming that the engine runs at an average speed of $$200$$ rpm, the polar mass moment of inertia ( in kg.m² ) of a flywheel to keep the speed fluctuation within $$\, \pm 0.5\% $$ of the average speed is _______ .

What is the natural frequency of the spring mass system shown below? The contact between the block and the inclined plane is frictionless. The mass of the block is denoted by $$m$$ and the spring constants are denoted by $${k_1}$$ and $${k_2}$$ as shown below.