An object with a mass of $3 k g$ $3 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x^{2} - 3 x + 6$ $u_k(x)= x^2-3x+6$ . How much work would it take to move the object over #x in [1, 3], where x is in meters?

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An object with a mass of $3 k g$ $3 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x^{2} - 3 x + 6$ $u_k(x)= x^2-3x+6$ . How much work would it take to move the object over #x in [1, 3], where x is in meters?

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Answer 1 · 2016-07-12T12:39:51

We know that the work done against variable frictional force is

$w = \int_{1}^{3} μ_{k} (x) m g d x$ $w=int_1^3mu_k(x)mgdx$

$= m g \int_{1}^{3} (x^{2} - 3 x + 6) d x$ $=mgint_1^3(x^2-3x+6)dx$

$= m g {[\frac{1}{3} \cdot x^{3} - \frac{3}{2} \cdot x^{2} + 6 x]}_{1}^{3}$ $=mg[1/3*x^3-3/2*x^2+6x]_1^3$

$= 3 \cdot 9.8 [(\frac{1}{3} \cdot 3^{3} - \frac{3}{2} \cdot 3^{2} + 6 \cdot 3) - (\frac{1}{3} \cdot 1^{3} - \frac{3}{2} \cdot 1^{2} + 6 \cdot 1)$ $=3*9.8[(1/3*3^3-3/2*3^2+6*3)-(1/3*1^3-3/2*1^2+6*1)$
$= 29.4 \cdot (13.5 - 4.8) = 255.78 J$ $=29.4*(13.5-4.8)=255.78J$

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An object with a mass of $3 k g$ $3 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x^{2} - 3 x + 6$ $u_k(x)= x^2-3x+6$ . How much work would it take to move the object over #x in [1, 3], where x is in meters?

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An object with a mass of 3kg3 kg is pushed along a linear path with a kinetic friction coefficient of uk(x)=x2−3x+6u_k(x)= x^2-3x+6 . How much work would it take to move the object over #x in [1, 3], where x is in meters?

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An object with a mass of $3 k g$ $3 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x^{2} - 3 x + 6$ $u_k(x)= x^2-3x+6$ . How much work would it take to move the object over #x in [1, 3], where x is in meters?