An object with a mass of $7 k g$ $7 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 3 + 2 sin (x)$ $u_k(x)= 3+2sin(x)$ . How much work would it take to move the object over #x in [pi, 4pi], where x is in meters?

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Answer 1 · 2017-10-21T06:14:08

The work is $= 1665 J$ $=1665J$

We need

$\int sin x d x = - cos x + C$ $intsinxdx=-cosx+C$

The work done is

$W = F \cdot d$ $W=F*d$

The frictional force is

$F_{r} = μ_{k} \cdot N$ $F_r=mu_k*N$

The normal force is $N = m g$ $N=mg$

The mass is $m = 7 k g$ $m=7kg$

$F_{r} = μ_{k} \cdot m g$ $F_r=mu_k*mg$

$= 7 \cdot (3 + 2 sin x) g$ $=7*(3+2sinx)g$

The work done is

$W = 7 g \int_{π}^{4 π} (3 + 2 sin x) d x$ $W=7gint_(pi)^(4pi)(3+2sinx)dx$

$= 7 g \cdot {[3 x - 2 cos x]}_{π}^{4 π}$ $=7g*[3x-2cosx]_(pi)^(4pi)$

$= 7 g (12 π - 2 cos (4 π)) - (3 π - 2 cos π)$ $=7g(12pi-2cos(4pi))-(3pi-2cospi)$

$= 7 g (9 π - 2 - 2)$ $=7g(9pi-2-2)$

$= 7 g (9 π - 4)$ $=7g(9pi-4)$

$= 1665 J$ $=1665J$

The value of $g = 9.8 m s^{- 2}$ $g=9.8ms^-2$

An object with a mass of 7kg7 kg is pushed along a linear path with a kinetic friction coefficient of uk(x)=3+2sin(x)u_k(x)= 3+2sin(x) . How much work would it take to move the object over #x in [pi, 4pi], where x is in meters?