A 1.0 -cm-diameter venturi flowmeter is inserted in a \(2.0-\mathrm{cm}-\) diameter pipe carrying water (density \(1000 \mathrm{kg} / \mathrm{m}^{3}\) ). Find (a) the flow speed in the pipe and (b) the volume flow rate if the pressure difference between venturi and unconstricted pipe is \(17 \mathrm{kPa}.\)

Firstly, use Bernoulli's equation which shows that increase in the speed of a fluid occurs simultaneously with decrease in its pressure or potential energy. The pressure difference given in the problem can be used to find the speed at the constricted part (venturi meter). Bernoulli's equation can be rearranged to isolate \(v_{2}\), the speed at the constricted part: \[v_{2} = \sqrt{{\frac{{2(p_{1} - p_{2})}}{{\rho}}}}\] Where, \(p_{1}\) and \(p_{2}\) are the pressures at the unconstricted and constricted parts of the pipe respectively, and \(\rho\) is the density of the liquid (water in this case). Substituting the given values we get: \[v_{2} = \sqrt{{\frac{{2(17000)}}{{1000}}}}\] which should be solved to find the value of \(v_{2}\)

Next, use the equation of continuity to find the flow speed in the pipe which states that the mass flow rate (ρAv) must be constant. Denote \(v_{1}\) as the speed of flow in the pipe and \(A_{1}\), \(A_{2}\) as the cross-sectional areas of the pipe and venturi meter respectively. According to the equation of continuity, the product of the cross-sectional area and the flow speed should be equal in both the pipe and venturi meter: \[v_{1} = \frac{{A_{2}}}{{A_{1}}} \cdot v_{2}\] The \(A_{1}\) and \(A_{2}\) can be written as πr² where r is the radius of pipe and venturi meter respectively. Substitute the given diameters to find the radii, which then allow to calculate the areas. After finding the areas, substitute the areas and \(v_{2}\) into the equation to find \(v_{1}\).

Now, volume flow rate (\(\dot{V}\)) can be computed using \(v_{1}\) and the cross-sectional area of the pipe (\(A_{1}\)). The equation to compute the volume flow rate is: \[\dot{V} = A_{1} \cdot v_{1}\] Using the calculated values for \(A_{1}\) and \(v_{1}\), substitute them into the equation to find the volume flow rate.

The final answer will be the flow speed in the pipe (part a) and the volume flow rate (part b) which were calculated in steps 2 and 3 respectively.