Question

An object with a mass of `1 kg` is pushed along a linear path with a kinetic friction coefficient of `u_k(x)= 3x^2+x+2`. How much work would it take to move the object over #x in [1, 3], where x is in meters?

Answer 1

`W=335,54J`

Explanation:

`F_f=u_k*N`
`F_f=" Friction force"`
`"N:Normal force to the contacting surfaces"`
`N=m*g " m:mass of object, g:acceleration of gravity"`
`F_f=u_k*m*g`
`W=int_1^3 F_f*d x`
`W:"work doing by Friction force"`
`W=int_1^3 u_k*m*g*d x`
`W=mg int_1^3 (3x^2+x+2)d x`
`W=mg[3*1/3x^3+1/2x^2+2x ]_1^3+C`
`W=1*9,81[x^3+1/2x^2+2x]_1^3 +C`
`u_k(0)=2 ; then C=2`
`W=9,81([27+4,5+6]-[1+0,5+2])+2`
`W=9,81(37,5-3,5)+2`
`W=9,81*34+2`
`W=335,54J`