An object with a mass of $6 k g$ $6 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 + csc x$ $u_k(x)= 2+cscx$ . How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?

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An object with a mass of $6 k g$ $6 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 + csc x$ $u_k(x)= 2+cscx$ . How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?

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Answer 1 · 2017-08-01T12:55:39

The work is $= 467 J$ $=467J$

Explanation:

We need

$\int csc x d x = ln ∣ ∣ tan (\frac{x}{2}) ∣ ∣$ $intcscxdx=ln|tan(x/2)|$

The work done is

$W = F \cdot d$ $W=F*d$

The frictional force is

$F_{r} = μ_{k} \cdot N$ $F_r=mu_k*N$

The normal force is $N = m g$ $N=mg$

The mass is $m = 6 k g$ $m=6kg$

$F_{r} = μ_{k} \cdot m g$ $F_r=mu_k*mg$

$= 6 (2 + csc x) g$ $=6(2+cscx)g$

The work done is

$W = 6 g \int_{\frac{1}{8} π}^{\frac{7}{8} π} (2 + csc x) d x$ $W=6gint_(1/8pi)^(7/8pi)(2+cscx)dx$

$= 6 g \cdot {[2 x + ln ∣ ∣ tan (\frac{x}{2}) ∣ ∣]}_{\frac{1}{8} π}^{\frac{7}{8} π}$ $=6g*[2x+ln|tan(x/2)|]_(1/8pi)^(7/8pi)$

$= 6 g ((2 \cdot \frac{7}{8} π + ln ∣ ∣ ∣ tan (\frac{7}{16} π) ∣ ∣ ∣) - (2 \cdot \frac{1}{8} π + ln ∣ ∣ ∣ tan (\frac{1}{16} π) ∣ ∣ ∣))$ $=6g((2*7/8pi+ln|tan(7/16pi)|)-(2*1/8pi+ln|tan(1/16pi)|))$

$= 6 g (\frac{7}{4} π - \frac{1}{4} π + 3.22)$ $=6g(7/4pi-1/4pi+3.22)$

$= 6 g (\frac{3}{2} π + 3.22)$ $=6g(3/2pi+3.22)$

$= 467 J$ $=467J$

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An object with a mass of $6 k g$ $6 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 + csc x$ $u_k(x)= 2+cscx$ . How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?

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Explanation:

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An object with a mass of 6kg6 kg is pushed along a linear path with a kinetic friction coefficient of uk(x)=2+cscxu_k(x)= 2+cscx . How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?

1 Answer

Explanation:

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An object with a mass of $6 k g$ $6 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 2 + csc x$ $u_k(x)= 2+cscx$ . How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?