## Using ratios to solve physics problems when variables change values

A common question when you learn an equation in physics is “how is the value of one variable affected when a second variable is changed?” For example, “An astronaut has a weight of 200 N as she stands on the surface of a certain moon of Jupiter. How much would she weigh standing on a second moon that has three times the mass and twice the radius of the first moon?”

Many students find this a difficult problem to calculate, but it is actually something that you can solve quickly with just a bit of thought. The trick is to recognize that problems of this sort can be solved with a simple ratio. Let’s see how to solve the problem posed above. We will call the astronaut’s weight on the first moon F1 and on the second moon F2. The masses of the two moons are M1 and M2 respectively and their radii are r1 and r2. The mass of the astronaut is MA. Then from the universal law of gravitation:

$F_1= \dfrac{GM_1M_A}{r_1^{\; 2}}$

and

$F_2= \dfrac{GM_2M_A}{r_2^{\; 2}}$

But we are told that the mass of the second moon is three times the mass of the first moon and its radius is twice the radius of the first moon. That means that M= 3 M1 and r= 2 r1. We substitute these expressions into the equation for F2 above. This gives

$F_2= \dfrac{G(3M_1)M_A}{({2r_1)}^2}= \dfrac{3GM_1M_A}{4r_1^{\; 2}}$

Now divide this equation by the first equation and all the variables cancel out:

$\dfrac{F_2}{F_1}=\dfrac{\dfrac{3GM_1M_A}{4r_1^{\; 2}}}{\dfrac{GM_1M_A}{r_1^{\; 2}}}=\dfrac{3}{4}$

$\therefore F_2= \dfrac{3}{4}F_1=150 \; N$

With some practice, you will find that you can do many of these problems in your head!

Physics
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