Answer
(a) The pressure of the fluid in the syringe must be $51.4~mm~Hg$
(b) A force of $0.685~N$ must be applied to the plunger.
Work Step by Step
(a) We can use Poiseuille's law to find an expression for the flow rate:
$Q = \frac{\pi~\Delta P~r^4}{8~\eta~L}$
$Q$ is the flow rate ($m^3~s^{-1}$)
$\Delta P$ is the pressure difference ($Pa$)
$r$ is the radius ($m$)
$\eta$ is the fluid viscosity
$L$ is the length of the tube ($m$)
We can find the required pressure difference $\Delta P$:
$\Delta P = \frac{8~Q~\eta~L}{\pi~r^4}$
$\Delta P = \frac{(8)~(2.50\times 10^{-7}~m^3/s)~(2.00\times 10^{-3}~Pa~s)~(0.030~m)}{(\pi)~(3.00\times 10^{-4}~m)^4}$
$\Delta P = 4715.7~Pa$
We can convert the pressure difference to units of mm Hg:
$\Delta P = (4715.7~Pa) \times \frac{1~mm~Hg}{133.3~Pa} = 35.4~mm~Hg$
Since the pressure difference is $35.4~mm~Hg$ and the pressure in the vein is $16.0~mm~Hg$, the pressure of the fluid in the syringe must be $51.4~mm~Hg$
(b) We can convert the pressure in the syringe to units of $Pa$:
$51.4~mm~Hg \times \frac{133.3~Pa}{1~mm~Hg} = 6851.6~Pa$
We can find the required force:
$F = P~A = (6851.6~N/m^2)(1.00\times 10^{-4}~m^2) = 0.685~N$
A force of $0.685~N$ must be applied to the plunger.
