For the epicyclic gear arrangement shown in the figure, $${{\omega _2} = 100}$$ rad/s clockwise (CW) and $${{\omega _{arm}} = 80}$$ rad/s counter clockwise (CCW). The angular velocity $${{\omega _5}}$$ (in rad/s) is

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

Match the items in columns $$I$$ and $$II$$

Column $$I$$
$$P.$$ Addendum
$$Q.$$ Instantaneous center of velocity
$$R.$$ Section modulus
$$S.$$ Prince circle

Column $$II$$
$$1.$$ Cam
$$2.$$ Beam
$$3.$$ Linkage
$$4.$$ Gear

A planetary gear train has four gears and one carrier. Angular velocities of the gears are $${\omega _1},{\omega _2},{\omega _3},$$ and $${\omega _4}$$ respectively. The carrier rotates with angular velocity $${\omega _5}.$$

What is the relation between the angular velocities of Gear $$1$$ and Gear $$4$$?