An object with a mass of $6 k g$ $6 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = 1 + 5 cot x$ $u_k(x)= 1+5cotx$ . How much work would it take to move the object over #x in [(11pi)/12, (5pi)/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 $u_{k} (x) = 1 + 5 cot x$ $u_k(x)= 1+5cotx$ . How much work would it take to move the object over #x in [(11pi)/12, (5pi)/8], where x is in meters?

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Answer 1 · 2017-04-02T09:36:55

The work is $= 320.5 J$ $=320.5J$

Explanation:

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

The nom'rmal reaction is $N = 6 g N$ $N=6gN$

The coefficient of kinetic friction is

$μ_{k} = \frac{F_{r}}{N}$ $mu_k=F_r/N$

$N = m g$ $N=mg$

So,

$F_{r} = μ_{k} \cdot N$ $F_r=mu_k*N$

$= (1 + 5 cot x) \cdot 6 g$ $=(1+5cotx)*6g$

The work done is

$W = F_{r} \cdot d$ $W=F_r*d$

$= \int_{\frac{11}{12} π}^{\frac{5}{8} π} (1 + 5 cot x) \cdot 6 g d x$ $=int_(11/12pi)^(5/8pi)(1+5cotx)*6gdx$

$= 6 g \int_{\frac{11}{12} π}^{\frac{5}{8} π} (1 + 5 cot x) d x$ $=6gint_(11/12pi)^(5/8pi)(1+5cotx)dx$

The integral of $cot x$ $cotx$ is

$= \int cot x d x = \int \frac{cos x d x}{sin x}$ $=intcotxdx=int(cosxdx)/sinx$

$= ln (| sin x |)$ $=ln(|sinx|)$

So,

$W = 6 g \cdot {[x + 5 ln (| sin x |)]}_{\frac{11}{12} π}^{\frac{5}{8} π}$ $W=6g*[x+5ln(|sinx|)]_(11/12pi)^(5/8pi)$

$= 6 g \cdot ((\frac{5}{8} π + 5 ln (∣ ∣ ∣ sin (\frac{5}{8} π) ∣ ∣ ∣) - (\frac{11}{12} π + 5 ln (∣ sin (\frac{11}{12} π ∣)))$ $=6g*((5/8pi+5ln(|sin(5/8pi)|)-(11/12pi+5ln(|sin(11/12pi|)))$

$= 6 g \cdot (- \frac{11}{12} π + \frac{5}{8} π + 5 (ln sin (\frac{5}{8} π) - ln sin (\frac{11}{12} π))$ $=6g*(-11/12pi+5/8pi+5(lnsin(5/8pi)-lnsin(11/12pi))$

$= 6 g \cdot 5.45$ $=6g*5.45$

$= 320.5 J$ $=320.5J$

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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 $u_{k} (x) = 1 + 5 cot x$ $u_k(x)= 1+5cotx$ . How much work would it take to move the object over #x in [(11pi)/12, (5pi)/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)=1+5cotxu_k(x)= 1+5cotx . How much work would it take to move the object over #x in [(11pi)/12, (5pi)/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 $u_{k} (x) = 1 + 5 cot x$ $u_k(x)= 1+5cotx$ . How much work would it take to move the object over #x in [(11pi)/12, (5pi)/8], where x is in meters?