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## 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/x2.

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 decreases by a factor of C2. If xdecreases by a factor of C, y increases by a factor of C2.

RATIOS For any two values of x—say, x1 and x2—we have Dividing the y1-equation by the y2-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/x2. 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