A particle moves along the x-axis under a force that varies with displacement according to F=ax2, where a is a positive constant and x is measured from the origin. The particle starts from x=0 and is pushed up to x=L.
a. (i) Sketch qualitatively the force–displacement graph for 0≤x≤L on properly labelled axes.
b. (ii) Treating the graph as a single geometric shape, derive an algebraic expression for the work done by the force in moving the particle from x=0 to x=L.
c. (iii) Using a=2Nm−2 and L=3m, evaluate the numerical value of the work obtained in part (ii).
d. (iv) Verify your result in part (iii) by performing the direct definite integral of F with respect to x over the stated limits.