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**HOW TO SIMPLIFY YOUR
COMPUTATIONAL LIFE**

Often in this course we will be interested
not in the absolute value of a quantity (e.g., that the distance of Jupiter from
the sun is 7.8 x 10^{8} km) but rather that the
quantity in question is so many itmes greater than some other quantity (that
Jupiter is 5.2 times as far away from the sun than is the earth).

**Another example:** if Circle A
has a radius of **2 m **and Circle B a radius of
**4 m**, then the circumference of B is how many
times larger than that of A? Or, more simply, what is the ratio of **C**_{B} to**
C**_{A}?
You could plug in the values **2 m **and **4 m** in the formula for the circumference, **C **=** 2***&pi *R,
multiply things out then take the ratio of the answers, but that is
unnecessarily cumbersome.

Instead, notice the following:** C**_{B }/ C_{A }=**
2***&pi *R_{B}
/ 2*&pi *R_{A} = **R**_{B} / R_{A}

So, the answer is immediately seen to be
**C**_{B
}/ C_{A }= **4/2 **=** 2 or
C**_{B }= **2 C**_{A}. That is, the circumference of a
circle is proportional to its radius, R; double R and you double C. The factor
'2* &pi*' is simply the constant of proportionality between C and
R.

**Similarly**, since the area of a circle is **A **=** &pi
R**^{2},
the ratio of the areas of the two circles is:

**A**_{B }/
A_{A }= **
***&pi *
(R_{B})^{2} / *&pi *
(R_{A})^{2}
=**
(R**_{B})^{2} / (R_{A})^{2} =
**(4/2)**^{2} = **4,
**and **A**_{B} = **4A**_{A
}

**So the area of a circle** is proportional to **R**^{2
}and *&pi
*is the constant of proportionality between the area and the
radius of a circle.

**The volume of a sphere** is given by **4/3***&pi *R^{2
}so the volume goes as the cube (third power) of
the radius and **4/3***&pi* is the constant of
proportionality. We can generalize these results and say that among similarly
shaped objects (squares or circles, etc.) linear distances on those objects
(e.g., the circumferences of cirlces) are directly proportional to the linear
dimension of the object, i.e., **C **is
proportional to **R**; the surface areas of
similar objects will be directly proportional to the square of their linear
dimenstions, **A **is proportional to **R**^{2}, and their volumes will be
directly proportional to the cube of their linear dimensions. Thus, if Jupiter
has **10 **times the radius of the earth (**R**_{Jupiter} =** 10R**_{Earth}), we can
say immediately that its circumference is also **10
C**_{Earth}, its surface area = **10**^{2}A_{Earth},
and its volume =**10**^{3}_{Earth}.

Sometimes quantities depend upon more than one factor, e.g.,
the force of gravity, **F** is proportional to
**(M**_{1} x** M**_{2})/R^{2}, is directly proportional to the product of the two masses involved,
and inversely proportional to the square of the distance between them. We say
'inversely' (and not 'directly') because here if you increase **R**, you decrease **F**; if
**R **is doubled, **F** is only **1/4** of its
original value; if **R **is tripled, **F **is **1/9** of its
original value, etc. If one of the masses is doubled, however, **F** is doubled; if both masses are doubled, **F **is increased by a factor of**
4**.

If we wanted to calculate the absolute value of the
gravitational force in a given situation, then we would have to use **F **= **G((M**_{1} x**
M**_{2})/R^{2}),
where **G** is the constant of proportionality
between **F **on the one hand, and **M** and **R **on the other.
As stated at the outset, however, generally we will be more interested in how
the force changes when M and **R **are changed, or
the ratio of the forces under different circumstances, rather than in the
absolute value of the force.