Question

A 100 g particle experiences the one-dimensional, conservative force Fx shown in (Figure 1). Let the zero of the potential energy be at x = 0 m . Figure1 of 1 The x component of force F subscript x is shown as a function of position x. The position is measured from 0 to 5 meters on the horizontal axis. The force is measured from negative 20 to 0 newtons on the vertical axis. The force component equals negative 20 newtons at the position interval from 0 to 3 meters, then increases linearly to 0 at 4 meters, and remains constant at the interval from 4 to 5 meters. Part A What is the potential energy at x = 1.0 m ? Hint: Use the definition of potential energy and the geometric interpretation of work.

Answer 1

The potential energy at x = 1.0 m is -20 J.

Step-by-step explanation:

To find the potential energy at x = 1.0 m, we need to integrate the force function from x = 0 to x = 1.0 m. Since the force is given as -20 N for the interval from 0 to 3 meters, we can write the force function as F(x) = -20 for 0 ≤ x ≤ 3. For the remaining intervals, the force is constant, either 0 or -20 N. We can integrate the force function to find the potential energy:

1. For 0 ≤ x ≤ 3: ∫ -20 dx = -20x
2. For 3 ≤ x ≤ 4: ∫ 0 dx = 0
3. For 4 ≤ x ≤ 5: ∫ -20 dx = -20x

The potential energy function U(x) is obtained by integrating the force function. Thus, U(x) = -20x for 0 ≤ x ≤ 3, and U(x) = -20x + C for 3 ≤ x ≤ 5, where C is the constant of integration. Since the potential energy is defined to be 0 at x = 0, we can find the constant of integration:

U(0) = 0 = -20(0) + C ⇒ C = 0

Therefore, the potential energy function is U(x) = -20x for 0 ≤ x ≤ 3, and U(x) = -20x for 3 ≤ x ≤ 5. To find the potential energy at x = 1.0 m, we evaluate the potential energy function at that point:

U(1.0) = -20(1.0) = -20 J