FIGURE EX10.28 shows the potential-energy diagram for a 500 g particle as it moves along the x-axis. Suppose the particle's mechanical energy is 12 J. (a) Where are the particle's turning points?

Verified step by step guidance

1. Identify the mechanical energy of the particle, which is given as 12 J.
2. Locate the points on the potential-energy diagram where the potential energy U(x) equals the mechanical energy (12 J). These points are the turning points.
3. From the graph, observe the x-axis values where the potential energy curve intersects the horizontal line at U = 12 J.
4. Note the x-coordinates of these intersection points. These coordinates represent the turning points of the particle.
5. Summarize the turning points by listing the x-coordinates where the potential energy equals the mechanical energy.

Key Concepts

Potential Energy

Potential energy (U) is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. In the context of the diagram, it represents the energy of the particle at various positions along the x-axis. The shape of the potential energy curve indicates how the energy changes with position, which is crucial for determining the particle's motion.

Mechanical Energy

Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE) in a system. For the particle in the question, the total mechanical energy is given as 12 J. This total energy remains constant if only conservative forces are acting, allowing us to find the turning points where the particle's kinetic energy is zero, and all mechanical energy is converted to potential energy.

Turning Points

Turning points occur where the particle changes direction, which happens when its kinetic energy is zero. At these points, the total mechanical energy equals the potential energy. By analyzing the potential energy diagram, we can identify the positions along the x-axis where the potential energy equals the mechanical energy of 12 J, indicating the locations of the turning points.