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

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

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Answer 1 · 2016-03-20T11:27:43

$797.1 J$ $797.1J$ rounded to one place of decimal

Explanation:

Force of kinetic friction which needs to be overcome to move the object

$F_{k} =$ $F_k=$ Coefficient of kinetic friction $μ_{k} \times$ $mu_ktimes$ normal force $η$ $eta$
where $η = m g$ $eta=mg$
In the problem $μ_{k}$ $mu_k$ is stated to be $u_{k} (x)$ $u_k(x)$
Inserting given quantities and taking the value of $g = 9.8 m / s^{2}$ $g=9.8 m//s^2$
$F_{k} = (2 x^{2} - 3 x) \times 8 \times 9.8 N$ $F_k=(2x^2-3x)times 8times 9.8 N$
$F_{k} = 78.4 (2 x^{2} - 3 x) N$ $F_k=78.4(2x^2-3x) N$

When this force moves through a small distance $d x$ $dx$ , the work done is given as
$F_{k} \cdot d x = [78.4 (2 x^{2} - 3 x)] \cdot d x$ $F_k cdotdx=[78.4(2x^2-3x)]cdot dx$
When the force moves through a distance from $x \in [2, 3]$ $x in [2, 3]$ , total work done is integral of RHS over the given interval.

Total work done $= \int_{2}^{3} 78.4 (2 x^{2} - 3 x) \cdot d x$ $=int_2^3 78.4(2x^2-3x)cdotdx$
$\Rightarrow$ $implies$ Total work done $= 78.4 \int_{2}^{3} (2 x^{2} - 3 x) \cdot d x$ $=78.4 int_2^3 (2x^2-3x)cdotdx$

$= 78.4 (2 \frac{x^{3}}{3} - 3 \frac{x^{2}}{2} + C) {∣ ∣ ∣}_{2}^{3}$ $=78.4 (2x^3 /3-3x^2/2+C)|_2^3$ , where C is constant of integration.
$=78.4 [ (2 3^3 /3-3 3^2/2+cancel C)-(2 2^3 /3-3 2^2/2+cancel C)]$
$=78.4 [ 18-27/2-16/3+6]$
$=78.4 [ (108-51-32+36)/6]$
$=78.4 [ 61/6]$
$797.1J$ rounded to one place of decimal

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

1 Answer

Explanation:

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An object with a mass of 8 kg8kg8 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 2x^2-3x uk(x)=2x2−3xu_k(x)= 2x^2-3x . How much work would it take to move the object over #x in [2, 3], where x is in meters?

1 Answer

Explanation:

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