Any help appreciated!
Pull out a blank piece of paper from your exam book. Hold it from one of its ends and
blow air horizontally over the top of the paper as shown in the figure below. If you blow
hard enough you will see the paper rise to become almost horizontal.
(a) Explain why this occurs.
(b) Estimate the air speed above the paper required to hold it horizontal if the paper
weighs m=4.
10-3 kg and has an area of A=0.04 m^2.
(The density of air is approximately ρ=1.2 kg/m3)
(a) The phenomenon in which the paper rises and becomes almost horizontal when air is blown over it is due to the Bernoulli's principle and the difference in pressure on both sides of the paper.
According to Bernoulli's principle, when the speed of a fluid (in this case, air) increases, its pressure decreases. When you blow air horizontally over the top of the paper, the air molecules acquire kinetic energy and move faster. This increased speed of the air over the top surface of the paper causes a decrease in pressure relative to the bottom surface where the air is relatively still.
The pressure difference between the top and bottom surfaces of the paper causes a net upward force on the paper, making it rise and become almost horizontal.
(b) To estimate the air speed required to hold the paper horizontal, we can use Bernoulli's principle and the equation for the lift force.
The lift force (F_lift) on the paper can be given by the equation:
F_lift = (ρ * A * v^2) / 2,
where ρ is the density of air, A is the area of the paper, and v is the velocity of the air.
In this case, the paper is nearly horizontal, so the lift force balances the weight of the paper:
F_lift = m * g,
where m is the mass of the paper and g is the acceleration due to gravity.
Setting these two equations equal to each other, we get:
(m * g) = (ρ * A * v^2) / 2.
Rearranging for v, we have:
v = sqrt((2 * m * g) / (ρ * A)).
Substituting the given values:
m = 4.10^-3 kg,
g ≈ 9.8 m/s^2,
A = 0.04 m^2,
ρ = 1.2 kg/m^3,
We can calculate the air speed (v) required to hold the paper horizontal as follows:
v = sqrt((2 * 4.10^-3 * 9.8) / (1.2 * 0.04)).
v = sqrt(0.064 / 0.048).
v ≈ sqrt(1.33).
Therefore, the estimated air speed required to hold the paper horizontal is approximately 1.15 m/s.