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

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Answer 1 · 2017-08-04T04:13:41

$W = 142$ $W = 142$ $J$ $"J"$

Explanation:

We're asked to find the necessary work that needs to be done on a $3$ $3$ - $kg$ $"kg"$ object to move on the position interval $x \in [2 l m, 3 l m]$ $x in [2color(white)(l)"m", 3color(white)(l)"m"]$ with a varying coefficient of kinetic friction $μ_{k}$ $mu_k$ represented by the equation

$ul(mu_k(x) = x^2-3x+6$

The work $W$ done by the necessary force is given by

$ulbar(|stackrel(" ")(" "W = int_(x_1)^(x_2)F_xdx" ")|)$ $color(white)(a)$ (one dimension)

where

$F_x$ is the magnitude of the necessary force
$x_1$ is the original position ( $2$ $"m"$ )
$x_2$ is the final position ( $3$ $"m"$ )

The necessary force would need to be equal to (or greater than, but we're looking for the minimum value) the retarding friction force, so

$F_x = f_k = mu_kn$

Since the surface is horizontal, $n = mg$ , so

$ul(F_x = mu_kmg$

The quantity $mg$ equals

$mg = (3color(white)(l)"kg")(9.81color(white)(l)"m/s"^2) = 29.43color(white)(l)"N"$

And we also plug in the above coefficient of kinetic friction equation:

$ul(F_x = (29.43color(white)(l)"N")(x^2-3x+6)$

Work is the integral of force, and we're measuring it from $x=2$ $"m"$ to $x=3$ $"m"$ , so we can write

$color(red)(W) = int_(2color(white)(l)"m")^(3color(white)(l)"m")(29.43color(white)(l)"N")(x^2-3x+6) dx = color(red)(ulbar(|stackrel(" ")(" "142color(white)(l)"J"" ")|)$

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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 $μ_{k} (x) = x^{2} - 3 x + 6$ $mu_k(x)= x^2-3x+6$ . How much work would it take to move the object over $x \in [2, 3]$ $x in [2, 3]$ , where $x$ $x$ is in meters?

1 Answer

Explanation:

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An object with a mass of 3 kg3kg3 kg is pushed along a linear path with a kinetic friction coefficient of mu_k(x)= x^2-3x+6 μk(x)=x2−3x+6mu_k(x)= x^2-3x+6 . How much work would it take to move the object over x in [2, 3]x∈[2,3]x in [2, 3], where xxx is in meters?

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Explanation:

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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 $μ_{k} (x) = x^{2} - 3 x + 6$ $mu_k(x)= x^2-3x+6$ . How much work would it take to move the object over $x \in [2, 3]$ $x in [2, 3]$ , where $x$ $x$ is in meters?