(II) The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero.
(a) In which interval(s) of 𝓍 is the force on the particle to the right?
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Verified step by step guidance

1. Identify the relationship between potential energy and force. Recall that the force exerted by a conservative field is the negative gradient of the potential energy, mathematically represented as F = -dU/dx.
2. Examine the potential energy curve for regions where the slope dU/dx is negative. In these regions, since the force is the negative of this slope, the force will be positive (to the right).
3. Look for intervals on the x-axis where the potential energy curve is decreasing as x increases. This indicates a negative slope.
4. Mark these intervals on the x-axis. These are the intervals where the force on the particle is directed to the right.
5. Summarize the intervals identified where the force is to the right, ensuring to specify the direction and the corresponding x-values.

Key Concepts

Potential Energy and Conservative Forces

Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic forces. A conservative force is one where the work done is independent of the path taken and depends only on the initial and final positions. In this context, the potential energy curve indicates how the potential energy changes with position, which directly influences the motion of the particle.

Total Energy and Motion

The total energy of a system is the sum of its kinetic and potential energy. When the total energy (E) is greater than the potential energy (U(x)), the particle has kinetic energy, meaning it is in motion. This relationship is crucial for determining the behavior of the particle, as it indicates that the particle cannot be at rest in regions where E > U(x).

Force and Potential Energy Relationship

The force acting on a particle can be derived from the potential energy curve using the relationship F = -dU/dx. This means that the force is related to the slope of the potential energy curve: if the slope is negative, the force acts in the positive direction (to the right), and if the slope is positive, the force acts in the negative direction (to the left). Analyzing the intervals of the potential energy curve helps identify where the force is directed.