In presence of friction, both the spring force and the frictional force act so as to oppose the compression of the spring as shown in Fig. 6.9.
We invoke the work-energy theorem, rather than the conservation of mechanical energy.
The change in kinetic energy is
`triangleK= K_(f)-K_(t)= 0-(1)/(2)mv^(2)`
The work done by the net force is
`W= -(1)/(2)kx_(m)^(2)- mu mg x_(m)`
Equatting we have
`(1)/(2)mv^(2)= (1)/(2)k x_(m)^(2)+ mu mg x_(m)`
Now `mu mg= 0.5xx10^(3)xx10= 5xx 10^(3)N` (taking `g= 10.0 ms^(-2)`). After rearranging the above equation we obtain the following quadratic equation in the unknown `x_(m)`.
`kx_(m)^(2)+ 2 mu mg x_(m)- mv^(2)= O`
`x_(m)=(-mu mg+[mu^(2)m^(2)g^(2)+mkv^(2)]^(1/2))/(k)`
where we take the positive square root since `x_(m)` is positive. Putting in numerical values we obtain `x_(m)= 1.35m`
which, as expected, is less than the result in Example 6.8.
If the two forces on the body consist of a conservative force `F_(c )` and a non-conservative force `F_(nc)`, the conservation of mechanical energy formula will have to be modified. By the WE theorem
`(F_(c )+F_(nc)) trianglex= triangleK`
But `F_(c ) triangle x= -triangle V`
Hence, `triangle(K+V)= F_(nc) triangle x`
`triangle E= F_(nc) trianglex`
where E is the total mechanical energy. Over the path this assumes the form
`E_(f)-E_(t)= W_(nc)`
where `W_(nc)` is the total work done by the non-conservative forces over the path. Note that unlike the conservative force, `W_(nc)` depends on the particular path i to f.
