A circular disk of radius $$'R'$$ rolls without slipping at a velocity $$v$$. The magnitude of the velocity at point $$'P'$$ (see figure) is

A straight rod length $$L(t),$$ hinged at one end freely extensible at the other end, roates through an angle $$\theta \left( t \right)$$ about the hinge. At time $$t,$$ $$L(t)=1m,$$ $$L(t)=1m/s,$$ $$\theta \left( t \right) = \pi /4$$ rad and $$\mathop \theta \limits^ \cdot \left( t \right) = 1\,\,rad/s.$$ The magnitude of the velocity at the other end of the rod is

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed $$V$$ towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.

The radial component of the fluid acceleration at $$r=R$$ is

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed $$V$$ towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.

The radial velocity $${V_r},$$ at any radius $$r$$, when the gap width is $$h,$$ is