The laplace transform of the periodic function $$f(t)$$ described by the curve below
$$i.e.\,\,f\left( t \right) = \left\{ {\matrix{ {\sin \,t,} & {if\left( {2n - 1} \right)\pi < t < 2n\pi \left( {n = 1,2,3,...} \right)} \cr 0 & {otherwise\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr } } \right.$$ is ___________.

The cylinder shown below rolls without slipping. In which direction does the friction force act? Towards which of the following points is the acceleration of the point of contact A on the cylinder directed?

A velocity field is given as $$$\overrightarrow V = 3{x^2}y\widehat i - 6xyz\widehat k$$$
Where $$x,y,z$$ are in $$m$$ and $$V$$ $$m/s.$$ Determine if

(i) It represents an incompressible flow
(ii) The flow is irrotational
(iii) The flow is steady .

Shown below are three pipe sections through which water flows as shown. Option $$(a)$$ to $$(d)$$ below pertain to the direction of the net force on the pipe section due to the flow of water.

State which of the options indicate the correct direction of force for pipe $$1,$$ pipe $$2$$ and pipe $$3.$$