- #1

Heisenberg7

- 73

- 13

- Homework Statement
- Figure 7-41 shows a cord attached to a cart that can slide along a frictionless horizontal rail aligned along an x axis. The left end of the cord is pulled over a pulley, of negligible mass and friction and at cord height h = 1.20 m, so the cart slides from x_1 = 3.00 m to x_2 =1.00 m. During the move, the tension in the cord is a constant 25.0 N. What is the change in the kinetic energy of the cart during the move? (Source: Fundamentals of Physics, Halliday and Resnick)

- Relevant Equations
- ##\Delta K = W##

We know that net work done is equal to the change in kinetic energy, so we write: $$\Delta K = W$$ The tension is acting at angle ##\theta## due to the x axis, so we will only be taking its x component ##T_x = T \cos{\theta}##. Since we can look at this as one dimensional motion (##T_y## does no work, only ##T_x## does; only tension does work), we can write: $$W = \int_{x_1}^{x_2} T_x \, dx \implies W = \int_{x_1}^{x_2} T \cos{\theta} \, dx \implies W = T \int_{x_1}^{x_2} \cos{\theta} \, dx$$ How does one go about solving this integral? I've seen someone's solution and it says that the work done by tension is ##W = Td## where ##d = \sqrt{x_1^2 + h^2} - \sqrt{x_2^2 + h^2}##. So, I guess my integral should be equal to this, but I have no idea how to solve it.