An object with a mass of $7 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_k(x)= 1+x-xcos(x/6)$ . How much work would it take to move the object over #x in [0, 4pi], where x is in meters?

Question

1407 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-25T10:13:49

The work is $=5501.7J$

We need to calculate the integral $intxcos(x/6)dx$

Perform this integration by parts

Let $u=x$ , $=>$ , $u'=1$

$v'=cos(x/6)$ , $=>$ , $v=6sin(x/6)$

$intuv'dx=uv-intu'vdx$

$intxcos(x/6)dx=6xsin(x/6)-6intsin(x/6)dx$

$=6xsin(x/6)+36cos(x/6)$

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 $m=7kg$

The acceleration due to gravity is $g=9.8ms^-2$

$F_r=mu_k*mg$

$=7*(1+x-xcos(x/6))g$

The work done is

$W=7gint_(0)^(4pi)(1+x-xcos(x/6))dx$

$=7g[x+1/2x^2-6xsin(x/6)-36cos(x/6)]_0^(4pi)$

$=7g((4pi+1/2(4pi)^2-24pisin(2/3pi)-36cos(2/3pi))-(0+0-0-36))$

$=7g(80.2)$

$=5501.7J$

An object with a mass of 7 kg7 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 1+x-xcos(x/6) u_k(x)= 1+x-xcos(x/6) . How much work would it take to move the object over #x in [0, 4pi], where x is in meters?