A pinion with radius $${r_1}$$, and inertia $${{\rm I}_1}$$ is driving a gear with radius $${r_2}$$ and inertia $${{\rm I}_2}$$ . Torque $${\tau _1}$$ is applied on pinion. The following are free body diagrams of pinion and gear showing important forces ($${F_1}$$and $${F_2}$$) of interaction. Which of the following relations hold true?

A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of point P is 10m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is ______

A mobile phone has a small motor with an eccentric mass used for vibrator mode. The location of the eccentric mass on motor with respect to center of gravity $$(CG)$$ of the mobile and the rest of the dimensions of the mobile phone are shown. The mobile is kept on a flat horizontal surface.

Given in addition that the eccentric mass = $$2$$ grams, eccentricity = $$2.19$$ mm, mass of the mobile = $$90$$ grams, g = $$9.81$$ $$m/{s^2}.$$ Uniform speed of the motor in $$RPM$$ for which the mobile will get just lifted off the ground at the end $$Q$$ is approximately

A precision instrument package (m = 1kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument. Assume that the isolation is designed with its natural frequency significantly lesser than 60Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is _____________.