A hypodermic syringe contains a medicine with the density of water (figure below). The barrel of the syringe has a cross-sectional area of 2.89 10-5 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force of magnitude 2.13 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.
Answer in m/s
C:\Users\Eddie\Documents\Karrine\physics\#24.gif
v2 = sqrt(F/A1*2/rho*(1-(A2/A1)^2))
= ~sqrt(F/A1*2/rho)
= 12.6 m/s
since A2/A1 =~0
To determine the medicine's flow speed through the needle, we can use Bernoulli's equation, which relates the pressure, velocity, and elevation between two points in a fluid flow. In this case, we can consider two points: the syringe barrel (Point 1) and the needle exit (Point 2).
Bernoulli's equation states: P1 + ρgh1 + (1/2)ρv1^2 = P2 + ρgh2 + (1/2)ρv2^2
Where:
P1 and P2 are the pressures at points 1 and 2 respectively.
ρ is the density of the fluid (given as the density of water).
g is the acceleration due to gravity.
h1 and h2 are the heights at points 1 and 2 respectively.
v1 and v2 are the velocities at points 1 and 2 respectively.
In this case, since the syringe is horizontal, we can assume that the heights (h1 and h2) are the same, and the acceleration due to gravity (g) does not affect the flow speed. So, the equation simplifies to:
P1 + (1/2)ρv1^2 = P2 + (1/2)ρv2^2
Given:
P1 = 1.00 atm (convert to Pa by multiplying by 1.013 x 10^5 Pa/atm)
P2 = 1.00 atm (same as P1)
v1 = 0 m/s (since there is no flow initially)
ρ = density of water = 1000 kg/m^3
v2 = ? (the flow speed we want to find)
Now, let's solve the equation for v2:
(1.013 x 10^5 Pa) + (1/2)(1000 kg/m^3)(0 m/s)^2 = (1.013 x 10^5 Pa) + (1/2)(1000 kg/m^3)v2^2
Simplifying the equation further:
(1.013 x 10^5 Pa) = (1.013 x 10^5 Pa) + (1/2)(1000 kg/m^3)v2^2
Rearranging the equation:
(1/2)(1000 kg/m^3)v2^2 = 0
Since the left-hand side of the equation is zero, it implies that (1/2)(1000 kg/m^3)v2^2 must also be zero.
Therefore, the flow speed of the medicine through the needle is 0 m/s.
Explanation: The given information indicates that a force of 2.13 N is exerted on the plunger, but it does not provide any information about the size of the opening or the resistance of the needle. Without this information, we cannot determine the flow speed of the medicine through the needle.