You need to fill a football with air to play with it. You know that your pump expels air at 10 ft/s. The needle of your pump has a radius of 5 micrometers. What is the volume flow rate of the air being pumped into the football?
To calculate the volume flow rate of the air being pumped into the football, we need to determine how much air is being pumped per unit time.
First, we need to find the cross-sectional area of the needle. The formula for the area of a circle is A = π * r^2, where A is the area and r is the radius. In this case, the radius is given as 5 micrometers, which is equivalent to 5 * 10^-3 millimeters or 5 * 10^-6 meters.
So, the cross-sectional area of the needle is A = π * (5 * 10^-6)^2 square meters.
Now, we can calculate the volume of air being pumped into the football per unit time. The volume flow rate (Q) is given by Q = A * v, where A is the cross-sectional area and v is the velocity or speed of air expelled from the pump.
In this case, the speed of air expelled from the pump is given as 10 ft/s. However, we need to convert it to meters per second to ensure consistent units. Since 1 foot is equivalent to 0.3048 meters, we have:
v = 10 ft/s * 0.3048 m/ft = 3.048 m/s.
Now, using the calculated cross-sectional area (A) and the velocity (v), we can determine the volume flow rate of air being pumped into the football.
Q = A * v = π * (5 * 10^-6)^2 * 3.048 cubic meters per second.
Calculating the above expression will give you the volume flow rate of the air being pumped into the football.