A 100 g particle experiences the one-dimensional, conservative force F_x shown in FIGURE P10.60. Let the zero of potential energy be at x = 0 m . What is the potential energy at x = 1.0, 2.0, 3.0, and 4.0 m? Hint: Use the definition of potential energy and the geometric interpretation of work.

Verified step by step guidance

1. Step 1: Recall the relationship between force and potential energy in one dimension: \( U(x) = - \int F_x \, dx \). The potential energy at a given position is determined by integrating the force over the distance from the reference point (where \( U(0) = 0 \)).
2. Step 2: Analyze the graph of \( F_x \). From \( x = 0 \) to \( x = 4 \), the force is constant at \( F_x = 10 \; \text{N} \). The work done (and thus the change in potential energy) is the area under the curve, which is a rectangle with height \( 10 \; \text{N} \) and width \( x \).
3. Step 3: From \( x = 4 \) to \( x = 8 \), the force decreases linearly from \( 10 \; \text{N} \) to \( 0 \; \text{N} \). The work done is the area of a triangle with base \( 4 \; \text{m} \) and height \( 10 \; \text{N} \). Use the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
4. Step 4: From \( x = 8 \) to \( x = 10 \), the force is zero, so no additional work is done, and the potential energy remains constant beyond \( x = 8 \).
5. Step 5: Calculate the potential energy at \( x = 1.0 \; \text{m}, \; x = 2.0 \; \text{m}, \; x = 3.0 \; \text{m}, \; \text{and} \; x = 4.0 \; \text{m} \) by summing the areas under the curve up to each position. For \( x > 4 \), include the triangular area contribution from \( x = 4 \) to \( x = 8 \).

Key Concepts

Conservative Forces

A conservative force is one where the work done by the force on an object moving between two points is independent of the path taken. This means that the work done can be fully recovered as potential energy. In this problem, the force Fx is conservative, allowing us to relate the force to potential energy through the work-energy principle.

Potential Energy (U)

Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. For conservative forces, potential energy can be calculated by integrating the force over a distance. In this case, we will calculate the potential energy at various positions by considering the area under the force versus position graph.

Work-Energy Theorem

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. For conservative forces, the work done can also be expressed as the negative change in potential energy. This relationship allows us to find potential energy values at different positions by calculating the work done by the force as the particle moves from the reference point.