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Consider Example 6.8 taking the coefficient of friction , mu to be and calculate the maximum compressionof the spring . |
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Answer» Solution :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. 
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