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Or phrased more abstractly, how to interpret the appearance of time dimension $[time]$ in the dimension of a physical quantity?

For example, the dimension of pressure is $[mass] [length]^{-1} [time]^{-2}$ corresponding to the SI-unit Pascal.

When you calculate pressure, you do not have to know any time. It is simply force per area – a division. There are many units that have time as component in them.

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Yes, but force is acceleration (times mass) so it needs time and has per second per second. – MBN Jan 26 '12 at 16:55

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Units don't always have intuitive meaning, particularly when you reformulate them in various ways. The pascal is intuitive when you express it as force/area, but you can reformulate that in dozens of ways that make no intuitive sense, such as mass*velocity*frequency/area.

In this case, the units of time get in there because they are in the units of force. It simply works out that way, there is no deeper meaning.

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good answer, but the example could be better - you could formulate pressure as mass of gas particle * mean velocity * collisions per second / area in which case your units are actually pretty intuitive. – user2963 Jan 26 '12 at 17:14
Hmm, good point. – Colin K Jan 26 '12 at 17:32

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