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hello friends, i am facing hard to understand the use of dimensional formula to derive relation between different physical quantities. Why can we say that proportionality constant k does not have any unit? -------------------------------------- in lecture 1 we found t=k(h/g)^1/2

OCW Scholar - Physics I: Classical Mechanics
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When we use dimensional analysis to guess/check a relation between quantities, we can express each quantity in terms of the basic SI units (from which other qties can be derived) and put this in the relation (if we have it). If we need to guess a relation we then check for the combination that fits best. But a quantity in our relation may be multiplied by a number. A number doesn't have units, so we cant use the technique to figure out what it must be. The analysis rests on the idea that in an (correct)equation the dimension of all quantities must be the same.
In the reletion you've quoted, we have no possible means to guess the constant which multiplies with (h/g)^1/2 to give the correct T. we must find that out from other means. Which is why we leave it as k. It means that, we know that it can have a constant multiple other than 1.
h is in units of distance and g is distance/time^2. Their ratio is time^2 and its square root is time units. Since the expression on the left of the equals sign is also time units, the units match if k has no units at all. Sometimes it helps to think of units like prime numbers; if you add two numbers that have the same prime factor, you can use the distributive law and it only appears once, but if you multiply or divide them, each unit either stays or cancels out. That's why the gravitational constant has the units N m^2/kg^2. If you cancel all the units on the right side of Newton's Law of Universal Gravitation, you get Newtons on the left. If you're crazy enough to use English units there will still be a conversion factor.

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Dimensional analysis is not what scientist use to derive formula. Deriving any formula would require detail experiments and consideration of physical significance of both equation and quantities involved in it. The reason the constant is 1 and unitless like in Newton's second law or others is that we know it before hand. Besides in my opinion it is also a type of convention.
thank you all very much.

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