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    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 108 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 CB to CA? 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: CB / CA = 2&pi RB / 2&pi RA = RB / RA

    So, the answer is immediately seen to be CB / CA = 4/2 = 2 or CB = 2 CA. 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 R2, the ratio of the areas of the two circles is:

    AB / AA = &pi (RB)2 / &pi (RA)2 = (RB)2 / (RA)2 = (4/2)2 = 4, and AB = 4AA

    So the area of a circle is proportional to R2 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 R2 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 R2, 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 (RJupiter = 10REarth), we can say immediately that its circumference is also 10 CEarth, its surface area = 102AEarth, and its volume =103Earth.

    Sometimes quantities depend upon more than one factor, e.g., the force of gravity, F is proportional to (M1 x M2)/R2, 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((M1 x M2)/R2), 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.