Question

The new Dyson® upright vacuum cleaner can create a pressure differential (gauge pressure) of -20 kPa at the tip of the 4-cm-diameter tool nozzle. (calculator permitted)

a. What percentage of atmospheric pressure can the vacuum’s motor remove?

b. What is the ultimate pick-up mass of this vacuum at the tool nozzle? (i.e. how massive of an object can the vacuum “pick up”?

c. The newer model Hoover® has a 6-cm-diameter tool nozzle. What gauge pressure must the vacuum supply at the nozzle tip to be able to perform equally well as the Dyson®?

Answer 1

Solution:

a) Gauge pressure = P1 = -20 kPa = -20 x 10^3 Pa

Atmospheric pressure = 1.03 x 10^5 Pa

Absolute pressure = gauge pressure + atmospheric pressure = -20000 + 1.03 x 10^5 Pa=81,000 Pa

% pressure that can be removed = (81,000) / (1.01 x 10^5) X 100 = 80.2%

b) Area = pi r^2 = pi (0.02)^2 = 0.0012566 m^2

Force = Pressure x Area = 20 x 10^3 x (0.0012566) = 25.13 N

Mass = 25.13/9.8 = 2.56 kg

c) Area (new) = pi (0.06/2)^2 = 0.00287 m^2

Force = mg = 25.13 N

Gauge Pressure = 25.13 / 0.00287 = 8888 Pa