## Math Box: Inverse-Square Relationships

**Learning Goal: **To understand and use inverse-square relationships.

Two quantities have an **inverse-square relationship** if y is inversely proportional to the *square* of x. We write the mathematical relationship as

y = A/x^{2}.

Here, *A* is a constant. This relationship is sometimes written as .

SCALING Inverse-square scaling means, for example:

- If you double x , you decrease y by a factor of 4.
- If you halve x , you increase y by a factor of 4.
- If you increase x by a factor of 3, you decrease y by a factor of 9.
- If you decrease x by a factor of 3, you increase y by a factor of 9.

Generally, if *increases* by a factor of C, y* decrease*s by a factor of C^{2}. If x*decreases* by a factor of C, y *increases* by a factor of C^{2}.

RATIOS For any two values of *x*—say, x_{1} and x_{2}—we have

Dividing the y_{1}-equation by the y_{2}-equation, we find

That is, the ratio of y-values is the inverse of the ratio of the squares of the corresponding values of x.

LIMITS As x becomes large, y becomes very small; as x becomes small, y becomes very large.

**A) Consider the case in which the constant A equals 16. Plot the graph of y=16/x**^{2}**.**

**B) ****Suppose the magnitude of the gravitational force between two spherical objects is 2000 N**** ****when they are ****100 km ****apart. What is the magnitude of the gravitational force Fg between the objects if the distance between them is 150 km**** ****?**

=> use the inverse square relationship

Fg = 889 N

**C) What is the gravitational force Fg between the two objects described in Part B if the distance between them is only 50 km?**

=> fg = 8000 N

**D) Which of the following describes how Fg and r are related in the previous two parts?**

=> As r shrinks to zero, Fg grows toward infinity. As r grows toward infinity, Fg shrinks to zero.

**E) Suppose that the magnitude of the force between two charged particles that are 2 cm apart is 50 N. What will the distance r between the particles be when the magnitude of the force between them is 200N?**

=> r= 1 cm

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