A system of particles in motion has mass center $$G$$ as shown in the figure. The particle $$i$$ has mass $${m_i}$$ and its position with respect to a fixed point $$O$$ is given by the position vector $${r_i}$$ . The position of the particle with respect to $$G$$ is given by the vector $${\rho _i}$$ . The time rate of change of the angular momentum of the system of particles about $$G$$ is (The quantity $$\mathop {{\rho _i}}\limits^{..} $$ indicates second derivative of $${\rho _i}$$ with respect to time and likewise for $${r _i}$$).

A point mass having mass $$M$$ is moving with a velocity $$V$$ at an angle $$\mathop \theta \limits^ \cdot $$ to the wall as shown in the figure. The mass undergoes a perfectly elastic collision with the smooth wall and rebounds. The total change (final minus initial) in the momentum of the mass is

Assuming constant temperature condition and air to be an ideal gas, the variation in atmospheric pressure with height calculated from fluid statics is

Consider fluid flow between two infinite horizontal plates which are parallel (the gap between them being $$50$$ $$mm$$). The top plate is sliding parallel to the stationary bottom plate at a speed of $$3$$ $$m/s.$$ The flow between the plates is solely due to the motion of the top plate. The force per unit area (magnitude) required to maintain the bottom plate stationary is _______________ $$N/{m^2}.$$

Viscosity of the fluid $$\mu = 0.44\,\,kg/m$$-$$s$$ and density $$\rho = 888$$ $$kg/{m^3}.$$