An object with a mass of $8 k g$ $8 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x + ln x$ $u_k(x)= x+lnx$ . How much work would it take to move the object over $x \in [2, 6]$ $x in [2, 6]$ , where $x$ $x$ is in meters?

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Answer 1 · 2017-10-17T06:54:55

The work is $= 1674.8 J$ $=1674.8J$

We need to compute

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

Apply the integration by parts

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

$v'=1$ , $=>$ , $v=x$

$I=uv-intu'vdx=xlnx-int1/x*xdx$

$=xlnx-int1dx=xlnx-x +C$

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 $m=8kg$

$F_r=mu_k*mg$

$=8*(x+lnx)g$

The work done is

$W=8gint_(2)^(6)(x+lnx)dx$

$=8g*[1/2x^2+xlnx-x]_(2)^(6)$

$=8g(1/2*36+6ln6-6)-(1/2*4+2ln2-2)$

$=8g(12+6ln6-2ln2)$

$=1674.8J$

The value of $g=9.8ms^-2$

An object with a mass of 8 kg8kg8 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= x+lnx uk(x)=x+lnxu_k(x)= x+lnx . How much work would it take to move the object over x in [2, 6]x∈[2,6]x in [2, 6], where xxx is in meters?