Question

Lou’s latest invention, aimed at urban dog owners, is the X-R-Leash. It is made of a rubber-like material that exerts a force Fx = (−5.7 N/m) x − (78 N/m2 ) x 2 when it is stretched a distance x. The ad claims, "You’ll never go back to your old dog leash after you’ve had the thrill of an X-R-Leash experience. And you’ll see a new look of respect in the eyes of your proud pooch." Find the work done on a dog by the leash if the person remains stationary, and the dog bounds off, stretching the X-R-Leash from x = 0 m to x = 22 m. Answer in units of kJ.

Answer 1

W = 29.06 KJ

Step-by-step explanation:

The work done while stretching the leash can be calculated by the following formula:

`W = \int\limits^b_a {F_(x)} \, dx \\`

whee,

W = Work Done = ?

Fₓ = Forcing Function = (-5.7 N/m)x - (78 N/m²)x²

a = starting point of x = 0 m

b = end point of x = 22 m

Therefore,

`W = \int\limits^(22\ m)_(0\ m) {(-(5.7\ N/m)x - (7.8\ N/m^(2))x^(2)}) \, dx \\Integrating\ we\ get:\\W = -((5.7\ N/m)x^(2))/(2) - ((7.8\ N/m^(2))x^(3))/(3)\\Applying\ limits:\\W = -((5.7\ N/m)(22\ m)^(2))/(2) - ((7.8\ N/m^(2))(22\ m)^(3))/(3) - 0\\W = - 1379.4\ J - 27684.8\ J\\W = 29064.2\ J`

W = 29.06 KJ