An object with a mass of #5 kg# is pushed along a linear path with a kinetic friction coefficient of #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?
1 Answer
Mar 13, 2016
Explanation:
The weight of the object is given by
The friction is given by
Consider the object being dragged/pushed along a small distance
The total work done to oppose the friction is given by
#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"#