Force & Newton's 2nd Law Problem

[crono_]

Brand new to the forum. My apologies if this problem is already posted on the site. I searched a bit but had no luck.

Homework Statement

Problem from Cutnell & Johnson PHYSICS 7th Edition. Chapter 4 - Problem #9

Two forces F_A and F_B are applied to an object whose mass is 8.0 kg. The larger force is F_A. When both forces point due east, the objects acceleration has a magnitude of 0.50 m/s². However, when F_A points due east and F_B points due west, the acceleration is 0.40 m/s², due east. Find (a) the magnitude of F_A and (b) the magnitude of F_B.

Homework Equations

F=ma

F_A + F_B = ma₁

F_A - F_B = ma₂

The Attempt at a Solution

I have the textbooks solution manual, but don't really understand the solution. Any help would be appreciated.

a) Adding equations 1 and 2 gives

F_A = m (a₁ + a₂) / 2

= (8.0 kg) (0.50 m/s² + 0.40 m/s²) / 2

= 3.6 N

b) Subtracting equation 2 from equation 1 gives

F_A = m (a₁ - a₂) / 2

= (8.0 kg) (0.50 m/s² + 0.40 m/s²) / 2

= 0.40 N

Ok, the final answers make sense when you plug the numbers into the equations. But I'm curious about why the equations are being added and subtracted from each other? In other words, how / why did the book mix the two equations together at get:

F_A = m (a₁ + a₂) / 2

F_A = m (a₁ - a₂) / 2

Again, probably just simple algebra that I'm looking over...

Thanks for any help!

 

[LowlyPion]

Welcome to PF.

It's a technique for eliminating variables, in this case they are trying to isolate Fa and Fb just in terms of the other variables and not themselves. It is only algebraic cleverness and not physics that calls for that approach. Other more brute force means, maybe not as elegant, could be employed.

 

[cepheid]

HOW did the textbook do it? Because if:

A = B​

and,

C = D​

then,

A + C = B + D​

This just follows from the fact that A and B are interchangeable...whenever you see one, you can substitute it for the other. Likewise for C and D.

I'm not sure if that qualifies as a formal proof, but I don't think you want to go so deep into the math as to question the nature of "equality" rather than merely accepting it and using it ;)

WHY did the textbook do this? Because it allows one to solve the problem.

 

[PhanthomJay]

You mean to say F_B = m(a1-a2)/2. Algebraically, when you add , subtract, multiply, divide, etc 2 equations together, you get a third equation which is equal to the first. Adding or subtracting eliminates one of the unknown variables, alowing you to solve for one of the unknowns.

 

[crono_]

Ok, I definitely need to brush up on the algebra. Thanks for the input everyone. I'm going to have to read and re-read your responses to fully understand what the textbook is doing.

Is anyone aware of where I could find information on this method?

Thanks again!