A triangular-shaped cantilever beam of uniform- thickness is shown in the figure. The young's modulus of the material of the beam is $$E$$. $$A$$ concentrated load $$P$$ is applied at the free end of the beam.

The maximum deflection of the beam is

A simple quick return mechanism is shown in the figure. The forward to return ratio of the quick return mechanism is 2 : 1. If the radius of the crank O₁ P is 125mm, then the distance 'd' (in mm) between the crank centre to lever pivot centre point should be

Match the approaches given below to perform stated kinematics/dynamics analysis of machine

Analysis
$$P.$$ Continuous relative rotation
$$Q.$$ Velocity and acceleration
$$R.$$ Mobility
$$S.$$ Dynamic - static analysis

Approach
$$1.$$ D' Alembert's principal
$$2.$$ Grubler's criterion
$$3.$$ Grashoff's law
$$4.$$ Kennedy's theorem

An epicyclic gear train is shown schematically in the adjacent figure The sun gear 2 on the input shaft is a 20 teeth external gear. The planet gear 3 is a 40 teeth external gear. The ring gear 5 is a 100 teeth internal gear. The ring gear 5 is fixed and the gear 2 is rotating at 60 rpm CCW (CCW $$=$$ counter-clockwise and CW $$=$$ clockwise)

The arm 4 attached to the output shaft will rotate at