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A robot arm that controls the position of a video camera in an automated surveillance system is manipulated by a servo motor that exerts a force on a push-rod. The force is given by F(x) = (12.9 N/m2)x2 ,
where x is the position of the end of the pushrod. If the push-rod moves from x1 = 1 m to x2 = 4 m, how much work did the servo motor do? Answer in units of J.

Respuesta :

Answer:

W = 270.9 J

Explanation:

given,  

F(x) = (12.9 N/m²) x²  

work = Force x displacement  

dW = F  .dx  

the push-rod moves from x₁= 1 m to x₂ = 4 m

integrating the above  

[tex]\int dW = \int_{x_1}^{x_2}F. dx[/tex]

[tex]W = \int_{x_1}^{x_2}F. dx[/tex]

[tex]W = \int_{1}^{4} (12.9 x^2) dx[/tex]

[tex]W = 12.9\times [\dfrac{x^3}{3}]_1^4 dx[/tex]

`[tex]W = 12.9\times [\dfrac{4^3}{3}-\dfrac{1^3}{3}] dx[/tex]

W = 270.9 J

work done by the motor is W = 270.9 J

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