An object with a mass of $8 k g$ $8 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x ln x$ $u_k(x)= xlnx$ . How much work would it take to move the object over #x in [2, 3], where x is in meters?

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Answer 1 · 2017-10-09T05:55:34

The work is $= 180.9 J$ $=180.9J$

We need to compute

$I = \int x ln x d x$ $I=intxlnxdx$

Apply the integration by parts

$u = ln x$ $u=lnx$ , $\Rightarrow$ $=>$ , $u'=1/x$

$v'=x$ , $=>$ , $v=x^2/2$

$I=uv-intu'vdx=1/2x^2lnx-int1/x*x^2/2dx$

$=1/2x^2lnx-1/2intxdx=1/2x^2lnx-1/2*x^2/2$

$=1/2x^2lnx-1/4x^2+C$

The work done is

$W=F*d$

The frictional force is

$F_r=mu_k*N$

The normal force is $N=mg$

The mass is $m=8kg$

$F_r=mu_k*mg$

$=8*(xlnx)g$

The work done is

$W=8gint_(2)^(3)(xlnx)dx$

$=8g*[1/2x^2lnx-1/4x^2]_(2)^(3)$

$=8g(1/2*9*ln3-1/4*9)-(1/2*4ln2-1/4*4)$

$=8g(9/2ln3-2ln2-5/4)$

$=180.9J$

An object with a mass of 8 kg8kg8 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= xlnx uk(x)=xlnxu_k(x)= xlnx . How much work would it take to move the object over #x in [2, 3], where x is in meters?