When you push down on the handle of a bicycle pump, a piston
in the pump cylinder compresses the air inside the cylinder.
When the pressure in the cylinder is greater than the pressure
inside the inner tube to which the pump is attached, air begins to
flow from the pump to the inner tube. As a biker slowly begins
to push down the handle of a bicycle pump, the pressure inside
the cylinder is 1.0 x 10^5 Pa, and the piston in the pump is 0.55 m
above the bottom of the cylinder. The pressure inside the inner
tube is 2.4 x 10^5 Pa. Calculate how far down must the biker push
the handle before air begins to flow from the pump to the inner
tube. Ignore the air in the hose connecting the pump to the inner
tube, and assume that the temperature of the air in the pump
cylinder does not change.
To solve this problem, we can use the principles of fluid mechanics and Boyle's law, which states that the pressure and volume of a gas are inversely proportional at constant temperature.
Here's how we can calculate the distance the biker must push the handle before air begins to flow from the pump to the inner tube:
Step 1: Determine the pressure difference between the cylinder and the inner tube.
The pressure difference is given by the equation:
ΔP = P_inner tube - P_cylinder
ΔP = 2.4 x 10^5 Pa - 1.0 x 10^5 Pa
ΔP = 1.4 x 10^5 Pa
Step 2: Determine the volume change of the cylinder.
Since the biker slowly pushes down the handle, the volume change of the cylinder can be approximated as the volume of the piston.
Volume_change = (area of the piston) * (distance the biker pushes the handle)
To calculate the area of the piston, we need to know its radius or diameter.
Step 3: Calculate the distance the biker must push the handle.
Using Boyle's law, we can relate the pressure difference to the volume change.
Boyle's law states: P1 * V1 = P2 * V2
where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
In our case, the initial pressure is P_cylinder = 1.0 x 10^5 Pa, and the final pressure is P_inner tube = 2.4 x 10^5 Pa.
The initial volume would be the volume of the cylinder before pushing down the handle, which can be found using the area and height of the cylinder.
Let's assume that the cylinder has a radius of 0.2 m (diameter = 0.4 m) and a height of H meters.
The area of the piston would be calculated as:
Area_piston = π * r^2,
where r is the radius of the piston.
Finally, we can substitute the values into the equation and solve for the distance the biker must push the handle:
P_cylinder * (π * r^2) * H = P_inner tube * (π * r^2) * (H + distance)
Simplifying, we get:
P_cylinder * H = P_inner tube * (H + distance)
Solving for the distance, we have:
distance = [(P_cylinder * H) - (P_inner tube * H)] / P_inner tube