Showing posts with label Strength of Materials. Show all posts
Showing posts with label Strength of Materials. Show all posts

## Sunday, January 6, 2013

### Ratio, Proportion - Elongation Problem

Category: Algebra, Strength of Materials

"Published in Newark, California, USA"

The elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area. A steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it. A certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs. Compute the minimum permissible area of the member.

Solution:

To illustrate the problem, let's draw the figure as follows

 Photo by Math Principles in Everyday Life

As you can see in the figure that when you applied a force F at the end of a metal with length L, there will be an elongation with length ∆L. From the first statement of the word problem, "the elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area," the working equation can be written as follows

and

Combining the two equations above, we have

or

where

∆L = elongation of a metal
k = proportionality constant
F = applied force
L = length of a metal
A = cross sectional area of a metal

From the second statement of a word problem, "a steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it," substitute the given items to the working equation in order to get the value of k as follows

or

From the third statement of a word problem, "a certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs," substitute the given items to the working equation in order to get the value of A as follows

Therefore, the minimum permissible area of the member is 0.1998 in2