A robot arm controls the position of a video camera and is manipulated by a motor that exerts a force on the arm. The varying force depends on the displacement of the robot arm and is given by the function F(x) = 2.0 + 133x2. If the arm moves from a displacement of 1.0 cm to 5.0 cm, how much work did the motor do?
To find the amount of work done by the motor, we need to calculate the area under the curve of the force function F(x) over the displacement interval from 1.0 cm to 5.0 cm. This can be done using integration:
W = ∫1.0cm5.0cm F(x) dx
W = ∫1.0cm5.0cm (2.0 + 133x^2) dx
W = [2x + 44.33x^3]1.0cm5.0cm
W = [(2(5) + 44.33(5)^3) - (2(1) + 44.33(1)^3)] J
W = 5,581.6 J
Therefore, the motor did 5,581.6 J of work to move the robot arm from 1.0 cm to 5.0 cm displacement.