Question

The amount of work done on an object is the force applied to the object times the distance the object is moved while the force is applied or, more succinctly, work=(force)(dis distance). We will use the International System of Units: Force is measured in Newtons, N, and distance is measured in meters , m. Weight is the gravitational force applied to the object and cable, therefore , weight = force How much work is done lifting a 1500 N object from the ground to the top of a 40 m building if the cable weighs 3 N per m ? Here the 1500 N object is constant , but as you retract the cable , the overall weight (force ) decreases . The diagram to the right can help you derive the integral to calculate the work done from the ground to the top of the buildingWrite the integral and solve.

Answer 1

Total work done = work done on object + work done on cable

From the information given,

weight of object = 1500 N

weight of rope = 3N per m

height interval = 0 to 40 m

Work = force x distance

If distance = x, then, we have the integral below

$\begin{gathered} \int_0^{40}(1500\text{ + \lparen40 - x\rparen3\rparen dx} \\ \int_0^{40}(1500\text{ + 120 - 3x\rparen dx} \\ \int_0^{40}(620\text{ - 3x\rparen dx} \\ 1620x\text{ - }\frac{3x^2}{2} \\ \end{gathered}$

By substituting x = 40 and x = 0, it becomes

1620(40) - 3/2 * (40)^2 - 1620(0) - 3/2 * (0)^2

= 64800 - 2400

= 62400 Nm

the work done from the ground to the top of the building is 62400 Nm