# An object with a mass of 8 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 5x^2-3x+1 . How much work would it take to move the object over x in [2, 3], where x is in meters?

Work $W = 1973.0 \overline{6} \text{ }$Joules

#### Explanation:

Work $\mathrm{dW} = F \cdot \mathrm{dx}$

$F = {\mu}_{k} \cdot N$

$F = {\mu}_{k} \cdot m \cdot g$

$\mathrm{dW} = {\mu}_{k} \cdot m \cdot g \cdot \mathrm{dx}$

$W = {\int}_{2}^{3} {\mu}_{k} \cdot m \cdot g \cdot \mathrm{dx}$

$W = {\int}_{2}^{3} \left(5 {x}^{2} - 3 x + 1\right) \cdot \left(8\right) \left(9.8\right) \mathrm{dx}$

$W = 78.4 {\int}_{2}^{3} \left(5 {x}^{2} - 3 x + 1\right) \mathrm{dx}$

W=78.4 [ (5x^3)/3-(3x^2)/2+x)]_2^3#

$W = 78.4 \left[45 - \frac{27}{2} + 3 - \left(\frac{40}{3} - 6 + 2\right)\right]$

$W = 1973.0 \overline{6}$

God bless....I hope the explanation is useful.