The force applied against an object moving horizontally on a linear path is described by #F(x)= cospix+3x #. By how much does the object's kinetic energy change as the object moves from # x in [ 1, 2 ]#?

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Answer 1 · 2016-02-09T18:33:54

#\Delta K = W = \int_1^2 F(x) dx = 9/2 # Joules

Work-Energy Theorem: The total work done by all the forces acting on a body must be equal to the change in its kinetic energy.

#\Delta K = K_f-K_i = W#

#W \equiv \int_{1}^{2} F(x) dx = \int_1^2 (cos\pix+3x) dx#,

#\qquad# #= 1/\pi\int_1^2 \frac{d}{dx}(sin\pi x) dx + [3/2 x^2]_1^2#

#\qquad# #=\frac{1}{pi} [sin \pix]_1^{2} + 9/2=0+9/2=4.5 J#