# What does it mean to be proportional to something?

I am asked a physics homework question, but it is really simply a mathematical question I think, dealing with proportional reasoning.

The period of a pendulum is proportional to the square root of its length.

A pendulum of length $2$ has a period of $3$.

They give units but it really should not matter here.

What I have done is written this down, and I am not really sure the first step is the correct one, because I am unsure how to read the question:

$$3P_1 = \sqrt 2$$

$$P_1 = \frac{\sqrt 2}{3}$$

They want to know about a pendulum of length $4.5$ so,

$$xP_2 = \sqrt {4.5}$$

My reasoning then is that:

$$\frac{\sqrt 2}{3} = \frac{\sqrt{4.5}}{x}$$

$$xP_2 = \frac{3\sqrt{4.5}}{\sqrt 2} = 4.5$$

I never really deal with proportional reasoning, so I am going out on a limb here to make this connection, but it all seems very intuitive. Of course, my intuition is not always right. My question then is, am I doing this right? Is this good reasoning?

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Hint:

Take T to be the time period of the pendulum.

$T_1 = k \cdot \sqrt {l_1}$

$T_2=k \cdot \sqrt{l_2}$

When you divide them both(Since none of the terms are $0$)

$\dfrac{T_1}{T_2}= \sqrt{\dfrac{l_1}{l_2}}$

Note: When L is proportional to M, you can write it as L=k. M, where k is a constant.

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Hm that is interesting, and it confirms my answer when I use the k, that really helps too. It is truly discomforting that my brain does all that without informing me how it was done.. Many thanks! – Leonardo Apr 7 '13 at 10:56
You're Welcome. And do note that there might be many cases of proportionality, when you come across Gravitational force (or maybe coulomb's force), you come across 'inversely proportional to'. – Inceptio Apr 7 '13 at 10:59
It may be noted that here the actual value of k is 2*pi/sqrt(g) where g is accelaration due to gravity. – Sreekanth Karumanaghat Apr 7 '13 at 11:24
@SreekanthKarumanaghat: That really doesn't matter when you have two relations, cause: they just scare off each other. – Inceptio Apr 7 '13 at 11:50
Yeah I know that I was just noting for info sake :) – Sreekanth Karumanaghat Apr 7 '13 at 12:08

The correct meaning of propotional:- If x is directly propotional to y. x/y = k(a constant)

Analytically

Graph between x and y is straight line whose slope is k.

Applying the above formula here. T1 = k sqrt(L1) T2 = k sqrt(L2)

dividing 2 by 1,we get T1/T2 = sqrt(L1)/sqrt(L2)

3/T2 = sqrt(2)/sqrt(4.5)

T2 = sqrt(4.5)*3/sqrt(2)

T2 = 4.5

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