Stuck on algebra to N2L solution

[NamaeKana]

I already understand the problem, but am having trouble with the algebra for textbook Fund of Physics Solutions, Halliday, 8e, Ch6 Prob 34. Copied here

http://i56.tinypic.com/20uwhw4.jpg

I've been staring at this solution to N2L for hours and can't figure out how to go from the top 4 N2L equations to the bottom equation for f (friction). I tried making both 'as' and 'ab' as 'a' but keep geting back to the orig equation.

Just give me a hint, please...

 

[NamaeKana]

o.k., i typed it here, but it's cryptic with (subscripts)

these four N2L equations:

-f = m(s) a(s)
F(ns) -F(Nb) -m(s)g = 0
f - F = m(b) a(b)
F(Nb) - m(b) g

are shown in the soln to resolve to

f = ( m(s) F ) / ( (m(s) + m(b) )

where do i begin ?

 

[SammyS]

How are the slab and block related?

 

[NamaeKana]

oh ! now i see. it's checking if the block on top is sliding on the slab. not deriving it.

 

[NamaeKana]

please view these images

here's the problem

http://i54.tinypic.com/23h6slh.jpg

the solution

http://i56.tinypic.com/2hhok79.jpg

 

[SammyS]

o.k., i typed it here, but it's cryptic with (subscripts)

these four N2L equations:

-f = m(s) a(s)
F(ns) -F(Nb) -m(s)g = 0
f - F = m(b) a(b)
F(Nb) - m(b) g

are shown in the sol'n to resolve to

f = ( m(s) F ) / ( (m(s) + m(b) )

where do i begin ?

What have you tried?

The solution says that a_s = a_b, they drop the subscripts & simply call them both "a".

Solve the 1st equation for a: $$a=\frac{-f}{m_s}\ .$$

Solve the 3rd equation for a: $$a=\frac{f-F}{m_b}\ .$$

Set the right hand sides equal to each other & solve for f .

Added in Edit:

It's probably easier Algebra-wise to do the following.

Solve the 1st equation for f:  f = -m_s·a

Plug this into the 3rd equation:  -m_s·a -F = m_b·a

Solve for a, then plug that back into the equation: f = -m_s·a

 

[NamaeKana]

thanks. i wouldn't have figured that out by myself in a millions years ;-). i get the logic, solve 1 & 3 for a, then take a and plug back into 1.