A particle moves along a line where the potential energy of its system depends on its position r as graphed in Figure P8.46. In the limit as r increases without bound, U(r) approaches +1 J. (a) Identify each equilibrium position for this particle. Indicate whether each is a point of stable, unstable or neutral equilibrium. (b) The particle will be bound if the total energy of the system is in what range? Now suppose that the system has energy −3 J. Determine (c) the range of positions where the particle can be found, (d) its maximum kinetic energy, (e) the location where it has maximum kinetic energy, and (f) the binding energy of the system— that is, the additional energy that it would have to be given in order for the particle to move out to r→ ∞ .

a) The equilibrium positions for this particle can be found at the positions where the slope of the potential energy graph (U(r)) is zero.

By looking at the graph, we can find three equilibrium positions at r ~ 0, r ~ 40, and r ~ 70. To determine whether each of these positions is stable, unstable, or neutral, we can examine the curvature of the graph around these positions.

- At r ~ 0, the graph is concave up, indicating that this is a point of stable equilibrium.
- At r ~ 40, the graph is concave down, indicating that this is a point of unstable equilibrium.
- At r ~ 70, the graph is concave up, indicating that this is a point of stable equilibrium.

b) The particle will be bound if the total energy of the system is less than +1 J, which is the potential energy for r → ∞. If the total energy were equal to or greater than +1 J, the particle could gain enough kinetic energy to move out to r → ∞.

c) With a total energy of -3 J, the particle can be found in positions where the potential energy is less than or equal to -3 J. Looking at the graph, this corresponds to the range of positions between r ~ 10 and r ~ 50.

d) The maximum kinetic energy will occur when the particle is at its lowest potential energy. This occurs at the bottom of the potential well, where U(r) ≈ -6 J. Since the total energy is conserved, at this point, the kinetic energy will be:

K_max = E_total - U(r) = -3 J - (-6 J) = 3 J

e) The particle has maximum kinetic energy when it is at the bottom of the potential well, which occurs at a location of r ~ 30.

f) The binding energy of the system is the additional energy needed to move the particle to r → ∞ . This can be found by subtracting the total energy (-3 J) from the potential energy at r → ∞ (+1 J):

binding energy = 1 J - (-3 J) = 4 J

a) Well, if the potential energy approaches +1 J as r increases without bound, we can conclude that the particle will have an equilibrium position at r = ∞. However, this is not stable equilibrium, but more like unstable equilibrium because the potential energy keeps increasing as r increases. So, the particle will just keep flying away in that case.

b) Now, if the total energy of the system is negative, which means it is less than -3 J, then the particle will be bound. It's like the particle is stuck in a not-so-fun amusement park ride that it can't escape from.

c) If the system has energy -3 J, then the particle can be found in a range of positions where the potential energy is less than -3 J. Unfortunately, I can't graph it from here, but let's call this position range "Clown Island". It's like a magical place where particles have a great time defying gravity!

d) The maximum kinetic energy of the particle can be found at the position where the potential energy is at its lowest. In other words, it's like when you finally get to the bottom of a roller coaster and you're screaming your lungs out. So, it will have its maximum kinetic energy at the bottom of Clown Island.

e) The location where the particle has maximum kinetic energy is, as mentioned earlier, at the bottom of Clown Island. It's like the perfect spot for the particle to party and let loose!

f) The binding energy of the system is the additional energy that would have to be given to the particle for it to escape to r→ ∞. But since the particle is already stuck in Clown Island, it can't really escape. So, I guess the binding energy is infinite! It's like trying to pry open a can of laughter, it's impossible!

Hope that clears things up, or at least makes you smile a bit!

(a) To identify the equilibrium positions, we need to locate the points where the potential energy curve crosses the x-axis. These points are where U(r) = 0 J.

From the information given, we know that as r increases without bound, U(r) approaches +1 J. So, there can be at most two equilibrium positions - one before the potential energy reaches its maximum and one after.

Let's call the equilibrium position before the maximum point "A" and the equilibrium position after the maximum point "B".

To determine the stability of these equilibrium positions, we need to analyze the slope of the potential energy curve at these points.

- If the slope is positive (ascending), it is an unstable equilibrium.
- If the slope is negative (descending), it is a stable equilibrium.
- If the slope is zero, it is a neutral equilibrium.

We will label the stability of the equilibrium positions once we analyze the potential energy curve.

(b) The particle will be bound if the total energy of the system is between the two extremes of the potential energy curve. In this case, the total energy must be greater than the minimum potential energy (U_min) and less than the maximum potential energy (U_max).

Let's denote the total energy as E. The particle will be bound if U_min < E < U_max.

(c) Given E = -3 J, we need to find the range of positions where the particle can be found.

First, let's sketch a rough graph of the potential energy curve based on the given information.

|
+1 J | . A
| .
| .
| .
| .
U (J) |
_______|____________________
| | | | | | | r

From the graph, we can observe that the particle can be found between the equilibrium positions A and B, since the potential energy in this range is below E = -3 J.

To find the range of positions numerically, we need more information about the potential energy function U(r).

(d) To determine the maximum kinetic energy, we need to find the point on the potential energy curve where the particle has the minimum potential energy (U_min). The difference in total energy between the maximum and minimum potential energy represents the maximum kinetic energy.

To find U_min, observe the lowest point on the potential energy curve, denoted as C.

|
+1 J | . A
| .
| .
| .
| .
U (J) | C
_______|____________________
| | | | | | | r

The maximum kinetic energy (K_max) is given by:

K_max = E - U_min

(e) The location where the particle has the maximum kinetic energy will be at the equilibrium position B, where U(r) = 0 J. At this position, all the energy is in the form of kinetic energy.

(f) The binding energy of the system is the additional energy needed for the particle to move out to r → ∞. This additional energy is equal to the difference in potential energy between the maximum point and the infinite separation limit.

Let's denote the potential energy at the maximum point as U_max and the potential energy at infinite separation as U_inf.

The binding energy (ΔE_bind) is given by:

ΔE_bind = U_inf - U_max

To calculate the range of positions, maximum kinetic energy, and binding energy accurately, we would need the expression or equation for the potential energy U(r) or more detailed information about the potential energy graph in Figure P8.46.

To answer the given questions, we need to analyze the potential energy graph (Figure P8.46) and understand the concepts of equilibrium, bound states, total energy, kinetic energy, and binding energy.

(a) Equilibrium positions:
- An equilibrium position is where the particle will come to rest and experience no net force or acceleration.
- In the potential energy graph, these positions correspond to the points where the slope of the curve is zero (i.e., where the derivative of U(r) with respect to r is zero).
- Identify these points on the graph and mark them as equilibrium positions.

Stable, Unstable, or Neutral equilibrium:
- Stable equilibrium: If a slight displacement from the equilibrium position results in a restoring force that brings the particle back towards the equilibrium position, it is stable.
- Unstable equilibrium: If a slight displacement from the equilibrium position results in a net force that pushes the particle further away from the equilibrium position, it is unstable.
- Neutral equilibrium: If a slight displacement from the equilibrium position neither brings the particle back nor pushes it away, it is neutral.
- Analyze the shape of the potential energy graph around each equilibrium position and determine whether it represents stable, unstable, or neutral equilibrium.

(b) Bound state:
- A particle is bound if the total energy of the system is negative.
- From the given information, we know that as r increases without bound, U(r) approaches +1 J.
- Therefore, the total energy (E) of the system needs to be less than +1 J (negative) for the particle to be bound. Identify the range of total energy values for which the system is bound.

(c) Range of positions:
- Given that the system has energy -3 J, we need to determine the range of positions where the particle can be found.
- The total energy (E) of a system is the sum of its kinetic energy (K) and potential energy (U). So, E = K + U.
- Rearrange the equation to find the range of positions where the particle can be found: K = E - U.
- Plug in the given values of E and U and calculate the range of positions.

(d) Maximum kinetic energy:
- The maximum kinetic energy occurs at the position where the potential energy is minimum.
- Look for the lowest point on the potential energy graph and find the corresponding position.
- Since kinetic energy is directly related to the velocity of the particle, the maximum kinetic energy occurs when the particle is at its fastest.

(e) Location of maximum kinetic energy:
- Determine the position on the graph where the particle has its maximum kinetic energy.
- Using the position identified in part (d), mark it as the location where the particle has the maximum kinetic energy.

(f) Binding energy:
- The binding energy of the system is the additional energy required to move the particle from its bound state to the point where r→ ∞ (approaching infinity).
- Calculate the binding energy by subtracting the potential energy at the bound state from the potential energy at r→ ∞ (which is +1 J according to the given information).

Remember to accurately read and interpret the potential energy graph to answer each part of the question.