A hypodermic needle is 2.98 cm in length and 0.28 mm in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water through it will be 1 g/s?

The equation you need to answer that question can be found at:

http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation

You will need to look up the viscosity of water. Assume room temperature.

To determine the pressure difference required for a specific flow rate through the needle, we can use Poiseuille's law, which relates flow rate to the dimensions of the needle and the pressure difference.

Poiseuille's law states that the flow rate (Q) through a needle is given by the equation:

Q = (π * ΔP * r^4) / (8 * η * L)

Where:
Q = Flow Rate
ΔP = Pressure Difference
r = Radius of the needle
η = Water viscosity
L = Length of the needle

Given:
Length of needle (L) = 2.98 cm = 0.0298 m
Diameter of needle (d) = 0.28 mm = 0.00028 m
Radius of needle (r) = d / 2 = 0.00014 m
Flow rate (Q) = 1 g/s = 0.001 kg/s
Water viscosity (η) = 0.001 Pa s (approximately)

Now we can calculate the pressure difference (ΔP):

ΔP = (Q * 8 * η * L) / (π * r^4)

Substituting the values:

ΔP = (0.001 kg/s * 8 * 0.001 Pa s * 0.0298 m) / (π * (0.00014 m)^4)

Calculating the pressure difference:

ΔP ≈ 84933 Pa

Therefore, a pressure difference of approximately 84933 Pa is required for a flow rate of 1 g/s through the given hypodermic needle.

To determine the pressure difference required for a specific flow rate through a hypodermic needle, we can make use of Poiseuille's Law, which relates the flow rate of a fluid to the pressure difference, viscosity of the fluid, length, and radius of the tube.

First, let's convert the given measurements to appropriate SI units:
Length of the needle = 2.98 cm = 0.0298 m
Diameter of the needle = 0.28 mm = 0.00028 m

Next, we need to calculate the radius of the needle:
Radius (r) = diameter / 2 = 0.00028 m / 2 = 0.00014 m

Now, let's assume the fluid flowing through the needle is water. The viscosity of water at room temperature (considered here) is approximately 0.001 Pa.s.

With these values, we can substitute them into Poiseuille's Law equation:

Flow rate (Q) = (π * r^4 * ΔP) / (8 * η * L)

where:
Q = flow rate (1 g/s = 0.001 kg/s)
r = radius of the needle
ΔP = pressure difference
η = viscosity of water
L = length of the needle

Rearranging the equation to solve for the pressure difference (ΔP):

ΔP = (8 * η * L * Q) / (π * r^4)

Substituting the known values:

ΔP = (8 * 0.001 kg/s * 0.0298 m) / (π * (0.00014 m)^4)

Calculating this expression will give us the required pressure difference to achieve a flow rate of 1 g/s through the needle.