A hypodermic syringe contains a medicine with the density of water (figure below). The barrel of the syringe has a cross-sectional area of 2.33 10-5 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force vector F of magnitude 1.99 N is exerted on the plunger, making medicine squirt from the needle. Determine the medicine's flow speed through the needle. Assume the pressure in the needle remains equal to 1.00 atm and that the syringe is horizontal.
To determine the flow speed of the medicine through the needle, we can use Bernoulli's equation, which relates the pressure, density, and velocity of fluid flow.
Bernoulli's equation is given by:
P1 + (1/2)ρV1^2 + ρgh1 = P2 + (1/2)ρV2^2 + ρgh2
In this case, we can assume that the height difference, h, between the points 1 and 2 is negligible (since the syringe is horizontal). Therefore, we can simplify the equation to:
P1 + (1/2)ρV1^2 = P2 + (1/2)ρV2^2
Where:
P1 = initial pressure (1.00 atm)
V1 = initial velocity (unknown)
P2 = final pressure (1.00 atm)
V2 = final velocity (unknown)
ρ = density of the medicine (same as water density)
Since the pressure remains constant inside the needle (1.00 atm), we can simplify further to:
(1/2)ρV1^2 = (1/2)ρV2^2
Now we can solve for the final velocity, V2:
V2^2 = V1^2
Taking the square root of both sides:
V2 = V1
So, the flow speed of the medicine through the needle (V2) is equal to the initial velocity (V1).
To find the initial velocity, we need to relate the force applied to the plunger with the pressure and cross-sectional area of the syringe. We can use the following equation:
F = P1 * A1
Where:
F = force applied on the plunger (1.99 N)
P1 = pressure inside the syringe (1.00 atm = 101325 Pa)
A1 = cross-sectional area of the syringe (2.33 x 10^-5 m^2)
Rearranging the equation to solve for the initial velocity, V1:
V1 = (F / (ρ*A1))^0.5
Substituting the given values, we get:
V1 = (1.99 N / (1000 kg/m^3 * 2.33 x 10^-5 m^2))^0.5
Simplifying the expression:
V1 = (1.99 N / (0.0233 kg/m^3))^0.5
V1 = (1.99 N / 0.0233 N/m^2)^0.5
V1 = (85.40 m^2/s^2)^0.5
V1 = 9.24 m/s
Therefore, the flow speed of the medicine through the needle is approximately 9.24 m/s.