# An object with a mass of 1 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 3x^2+12 . How much work would it take to move the object over #x in [1, 3], where x is in meters?

Work $W = 490 \text{ }$Joules

#### Explanation:

Given:
mass $m = 1 \text{ ""kg}$
Variable kinetic friction coefficient ${\mu}_{k} \left(x\right) = 3 {x}^{2} + 12$

Let $x$ the distance moved by the object

Let ${F}_{k}$ the force required to move the object
Let ${F}_{n}$ the normal force exerted by the surface on the object
${F}_{n} = m \cdot g$

${F}_{k} = {\mu}_{k} \cdot {F}_{n}$
${F}_{k} = \left(3 {x}^{2} + 12\right) \cdot m \cdot g$

The amount of work

$\mathrm{dW} = {F}_{k} \cdot \mathrm{dx}$

$W = {\int}_{1}^{3} {F}_{k} \cdot \mathrm{dx}$

$W = {\int}_{1}^{3} \left(3 {x}^{2} + 12\right) \cdot m \cdot g \cdot \mathrm{dx}$

$W = m g \cdot {\int}_{1}^{3} \left(3 {x}^{2} + 12\right) \cdot \mathrm{dx}$

$W = m g {\left[\left({x}^{3} + 12 x\right)\right]}_{1}^{3}$

$W = 50 \cdot m g$

$W = 50 \cdot \left(1\right) \cdot \left(9.8\right)$

$W = 490 \text{ }$Joules

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