## 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**** ****?**

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

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

**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

## Answer ( 1 )

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 Nwhen they are100 kmapart. 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