Due to the pressure of the air in a tire it exerts a certain force on the surfac

Question

Due to the pressure of the air in a tire it exerts a certain force on the surface area of the valve stem on the tire. In return, the valve stem must exert a certain force on the bit of air within that is in contact with it to counteract the pressure in the tire and prevent any of that air from leaking out.

How is the magnitude of that force related to the tire pressure and the cross sectional area of the valve stem?

If the diameter of the valve stem were doubled, by how much would the force exerted by the valve stem increase? [Hint: how does the area of a circle change when you double its diameter.]

Explanation / Answer

The force will be given by tire pressure x cross sectional area of valve

i.e. F= P x A

On doubling the diameter,

as Area= Pi x D^2 /4

therefore, Area= pi x (2D) ^2/4 = Pi x D^2 i.e. 4 times the original area

As, area becomes 4 times and pressure remains same

Thus, Force= P x 4A= 4times original force

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