An object with a mass of $1 k g$ $1 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 e^{x} - x + 3$ $u_k(x)= 2e^x-x+3$ . How much work would it take to move the object over $x \in [1, 4]$ $x in [1, 4]$ , where x is in meters?

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An object with a mass of $1 k g$ $1 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 e^{x} - x + 3$ $u_k(x)= 2e^x-x+3$ . How much work would it take to move the object over $x \in [1, 4]$ $x in [1, 4]$ , where x is in meters?

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Answer 1 · 2017-08-14T20:32:02

$W = 1033$ $W = 1033$ $J$ $"J"$

Explanation:

We're asked to find the work necessary to push an object on a certain position interval with a varying coefficient of kinetic friction.

The equation for work with a varying force is

$W = \int_{x_{1}}^{x_{2}} F_{x} l d x$ $W = int_(x_1)^(x_2) F_xcolor(white)(l)dx$

where

$x_{1}$ $x_1$ and $x_{2}$ $x_2$ are the initial and final positions
$F_{x}$ $F_x$ is the force, which will be equal to the friction force acting (but in the opposite direction):

$F_{x} = μ_{k} n = μ_{k} m g$ $F_x = mu_kn = mu_kmg$

So

$W = \int_{x_{1}}^{x_{2}} μ_{k} m g l d x$ $W = int_(x_1)^(x_2) mu_kmgcolor(white)(l)dx$

We know:

$x_{1} = 1$ $x_1 = 1$ $m$ $"m"$
$x_{2} = 4$ $x_2 = 4$ $m$ $"m"$
$μ_{k} = 2 e^{x} - x + 3$ $mu_k = 2e^x - x + 3$
$m = 1$ $m = 1$ $kg$ $"kg"$
$g = 9.81$ $g = 9.81$ ${m/s}^{2}$ $"m/s"^2$

Plugging these in:

$W = int_(1color(white)(l)"m")^(4color(white)(l)"m")(1color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)(2e^x-x+3)color(white)(l)dx = color(red)(ulbar(|stackrel(" ")(" "1033color(white)(l)"J"" ")|)$

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An object with a mass of $1 k g$ $1 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 e^{x} - x + 3$ $u_k(x)= 2e^x-x+3$ . How much work would it take to move the object over $x \in [1, 4]$ $x in [1, 4]$ , where x is in meters?

1 Answer

Explanation:

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An object with a mass of 1 kg1kg1 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 2e^x-x+3 uk(x)=2ex−x+3u_k(x)= 2e^x-x+3 . How much work would it take to move the object over x in [1, 4]x∈[1,4]x in [1, 4], where x is in meters?

1 Answer

Explanation:

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An object with a mass of $1 k g$ $1 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 e^{x} - x + 3$ $u_k(x)= 2e^x-x+3$ . How much work would it take to move the object over $x \in [1, 4]$ $x in [1, 4]$ , where x is in meters?