Question

A pump is being used to inflate a bicycle tyre. The radius of the wheel to the centre of the tyre is 30 cm, and the mean radius of the tyre is 1.5 cm. 

The tyre starts out completely deflated (i.e. there is no air in it). A pump is being used to inflate it which has a stroke volume of 8 × 10^-5 m3. The final pressure inside the tyre is to be 2 × 10^5 Pa, and the air pressure is 1.01 × 10^5 Pa. Calculate the number of strokes that will be needed. Assume that this will be done slowly to avoid heating up the air.

Answer 1

Find how many strokes will be needed to inflate the tire to the atmospheric pressure. First, determine its volume:

$$V=AL=\pi r^2·2\pi R=2\pi^2 r^2R.$$

The number of strokes:

$$N=\frac Vv=\frac{2\pi^2r^2R}{v}=16.65.$$

At this point, the unloaded tire reaches its maximum volume and then all air will increase the pressure, not volume. For the current state, we have

$$p_0V=nRT,$$

for the tire inflated to 2 atm:

$$pV=n'RT=KnRT,$$

divide one by another:

$$\frac{p_0}{p}=\frac{n}{Kn},\\\space\\ K=\frac{p}{p_0}=1.98.$$

The total number of strokes:

$$M=N+K=16.65+1.98=19.$$