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A hypodermic syringe contains a medicine with the density of water (figure below). The barrel of the syringe has a cross-sectional area of 2.88 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 2.01 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.
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To determine the medicine's flow speed through the needle, we can use Bernoulli's equation, which relates the pressure, velocity, and height of a liquid.
The equation is written as:
P1 + 0.5ρv1^2 + ρgh1 = P2 + 0.5ρv2^2 + ρgh2
Where:
P1 and P2 are the pressures at two points in the fluid (in this case, the needle and the syringe barrel),
ρ is the density of the fluid (since it has the density of water, we can use the density of water, which is 1000 kg/m^3),
v1 and v2 are the velocities at the two points,
g is the acceleration due to gravity (approximated as 9.8 m/s^2), and
h1 and h2 are the heights at the two points.
In this case, the heights h1 and h2 are the same since the syringe is horizontal and the needle is at the same height as the syringe barrel. Therefore, we can simplify the equation to:
P1 + 0.5ρv1^2 = P2 + 0.5ρv2^2
Since the pressure at both points is 1.00 atm, we can substitute the value into the equation:
1.00 atm + 0.5 * (1000 kg/m^3) * (v1^2) = 1.00 atm + 0.5 * (1000 kg/m^3) * (v2^2)
The pressure terms cancel out since they are equal, leaving us with:
0.5 * (1000 kg/m^3) * (v1^2) = 0.5 * (1000 kg/m^3) * (v2^2)
Now, let's solve for v2, the flow speed through the needle.
Dividing both sides of the equation by 0.5 * (1000 kg/m^3), we get:
v1^2 = v2^2
To solve for v2, take the square root of both sides, considering that the flow speed should be positive:
v2 = sqrt(v1^2)
Now, we need to find v1, the velocity at the syringe barrel. We can determine this using the force exerted on the plunger.
The force exerted on the plunger is given as 2.01 N. Force is equal to the product of pressure and area, so we can write the equation as:
F = P1 * A
Where:
F is the force (2.01 N),
P1 is the pressure (1.00 atm = 101325 Pa),
A is the cross-sectional area of the syringe barrel (2.88 * 10^-5 m^2).
Solving for P1, we have:
P1 = F / A
Substituting the values, we find:
P1 = (2.01 N) / (2.88 * 10^-5 m^2)
Now that we have P1, we can substitute it into the equation for v2 to find the flow speed through the needle.
v2 = sqrt(v1^2) = sqrt(P1 * (2 / ρ))
Finally, substitute the given density of water (1000 kg/m^3) into the equation, and calculate v2.