In an epicyclic gear train, shown in the figure, the outer ring gear is fixed, while the sun gear rotates counterclockwise at 100 rpm. Let the number of teeth on the sun, planet and outer gears to be 50, 25 and 100, respectively. The ratio of magnitudes of angular velocity of the planet gear to the angular velocity of the carrier arm is _________.

The damping ratio for a viscously damped spring mass system, governed by the relationship $$\,m{{{d^2}x} \over {d{t^2}}} + c{{dx} \over {dt}} + kx = f\left( t \right),\,\,\,$$ is given by

A thin uniform rigid bar of length $$L$$ and mass $$M$$ is hinged at point $$O,$$ located at a distance of $${L \over 3}$$ from one of its ends. The bar is further supported using springs, each of stiffness $$k,$$ located at the two ends. A particle of mass $$m = {m \over 4}$$ is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about $$O,$$ the natural frequency of the systems is

The molar specific heat at constant volume of an ideal gas is equal to $$2.5$$ times the universal gas constant ($$8.314$$ $$J/mol.K$$). When the temperature increases by $$100$$ $$K$$, the change in molar specific enthalpy is _______________ $$J/mol.$$