An elevator (lift) consists of the elevator cage and a counter weight, of mass m each. The cage and the counterweight are connected by a chain that passes over a pulley. The pulley is coupled to a motor. It is desired that the elevator should have a maximum stopping time of t seconds from a peak speed $$V.$$ If the inertias of the pulley and the chain are neglected, the minimum power that the motor must have is

A shell is fired from a cannon. At the instant the shell is just about to leave the barrel, its velocity relative to the barrel is 3m/s, while the barrel is swinging upwards with a constant angular velocity of 2 rad/sec. The magnitude of the absolute velocity of the shell is

The time variation of the position of a particle in rectilinear motion is given by $$x = 2{t^3} + {t^2} + 2t$$. If 'V' is the velocity and 'a' acceleration of the particle in consistent units, the motion started with

A simple pendulum of length 5m, with a bob of mass 1 kg, is in simple harmonic motion. As it passes through its mean position, the bob has a speed of 5m/s. the net force on the bob at the mean position is