Dimensions can go a long way in helping us get a useful formula. The two sides o

Question

Dimensions can go a long way in helping us get a useful formula. The
two sides of any physically meaningful equation must be
dimensionally as well as numerically equal. To get an equation for
the tension in the wall of a cylinder in terms of the pressure inside, we
first observe that
[ Tension ] =force / length
and [ Pressure ] = force/(length)2
So we might guess that (Tension) = (constant) x (Radius) x (Pressure)
Does this formula explain why
(a) blowing a balloon is hard to start ?
(b) arterioles get by with thinner walls than arteries under the same pressure?

Explanation / Answer

A)

balloon initially at its neutral position, or at 'zero' position. At this stage, chain molecules in an elastic materials are all tangled up, a mess kind of situation. Hence you will need somewhat higher force to blow up right at the start as we are starting to untangle the chain molecules. After initial stretching (blowing), elastic chain molecules have started untangling, situation is less mess-like, more orderly and you will require lesser force to continue blowing the balloon as molecules keep untangling. After some time balloon will be fully blown and its chain molecules are all straightened up, completely untangled.

B)

The arteries branch into smaller vessels, eventually becoming very small vessels called arterioles. Because of very small vessels, to sustain the blood pressure the walls of arterioles will become thinner.

Here between arterioles and arteries pressure is same but radipus of vessels is different, that cause the difference in thickness.

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