An object with a mass of $5 k g$ $5 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x + x sin x$ $u_k(x)= x+xsinx$ . How much work would it take to move the object over #x in [0, 5pi], where x is in meters?

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An object with a mass of $5 k g$ $5 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x + x sin x$ $u_k(x)= x+xsinx$ . How much work would it take to move the object over #x in [0, 5pi], where x is in meters?

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Answer 1 · 2016-03-13T03:54:42

$7 kJ$ $7 "kJ"$

Explanation:

The weight of the object is given by $W = m \cdot g$ $W = m*g$ . Since the object is not accelerating in the vertical direction, the normal force, $N$ $N$ , is exactly equal to the weight.

The friction is given by $F = u_{k} \cdot N$ $F = u_"k" * N$ .

Consider the object being dragged/pushed along a small distance $d x$ $"d"x$ . The amount of work needed is $F \cdot d x = N u_{k} (x) d x$ $F * "d"x = N u_k(x) "d"x$ .

The total work done to oppose the friction is given by

$\int_{0}^{5 π} F d x = N \int_{0}^{5 π} u_{k} (x) d x$ $int_0^{5pi} F dx = N int_0^{5pi} u_k(x) "d"x$

$= mg int_0^{5pi} (x+xsin(x)) "d"x qquad$ (integrate by parts)

$= mg [x^2/2 - xcos(x) + sin(x)]_0^{5pi}$

$= mg (frac{25pi^2}{2} + 5pi)$

$= (5 "kg") (9.81 "m/s"^2) (frac{25pi^2}{2} + 5pi)$

$= 6821 "J"$

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An object with a mass of $5 k g$ $5 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x + x sin x$ $u_k(x)= x+xsinx$ . How much work would it take to move the object over #x in [0, 5pi], where x is in meters?

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Explanation:

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An object with a mass of 5 kg5kg5 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= x+xsinx uk(x)=x+xsinxu_k(x)= x+xsinx . How much work would it take to move the object over #x in [0, 5pi], where x is in meters?

1 Answer

Explanation:

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An object with a mass of $5 k g$ $5 kg$ is pushed along a linear path with a kinetic friction coefficient of $u_{k} (x) = x + x sin x$ $u_k(x)= x+xsinx$ . How much work would it take to move the object over #x in [0, 5pi], where x is in meters?