One-Dimensional and Two-Dimensional Potentials

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Questions and Answers

What is the value of the second derivative $\frac{d^2V}{dx^2}$ at the point $x_1$?

0
+1
-4
-1 (correct)

At which points does the potential energy function $V(x)$ have minima?

$x_1$ and $x_4$
$x_2$ and $x_3$
$x_1$ and $x_3$
$x_4$ and $x_5$ (correct)

What are the turning points for the motion when $E = \frac{25}{54}$ J?

$x = 1$ and $x = 5$
$x = -1$ and $x = 3$ (correct)
$x = -2$ and $x = 4$
$x = 0$ and $x = 2$

What is the limit of the potential energy function as $x$ approaches ±∞?

$- fty$ (D) Signup and view all the answers

Which equation represents the first derivative of the potential energy function $V(x)$?

$\frac{dV}{dx} = -12x^{5} + 48x^3 - 24x$ (A) Signup and view all the answers

What does the energy conservation theorem imply for one-dimensional motion under an external potential?

The total energy is constant throughout the motion. (D) Signup and view all the answers

What does the second derivative of the potential energy function, $V(x)$, at a point indicate?

The nature of the point (min or max). (B) Signup and view all the answers

At which limit does the potential energy $V(x)$ approach negative infinity?

$ o ext{±} ext{infinity}$. (C) Signup and view all the answers

What characterizes the turning points in a one-dimensional potential energy function?

They correspond to points where the derivative of potential energy is zero. (A) Signup and view all the answers

What is the function of the angular momentum in the context of a central potential?

It is conserved if no external forces act on the system. (D) Signup and view all the answers

In one-dimensional motion, what calculation allows you to express the potential energy as a function of position?

Total energy minus kinetic energy. (C) Signup and view all the answers

What type of coordinate system is typically used for analyzing central potentials?

Polar coordinates, for ease of analysis. (B) Signup and view all the answers

Which scenario correctly describes the limits of potential energy $V(r)$ at infinity for central potentials?

Potential energy approaches negative infinity. (A) Signup and view all the answers

What does the expression for angular momentum $L = mr^2 heta$ imply about its dependence on the radius and angular velocity?

It increases with increasing radius for constant mass and angular velocity. (C) Signup and view all the answers

Given that the total energy of the system is defined as $E=rac{1}{2}m{rac{dr}{dt}}^2 + V_{eff}(r)$, what does $V_{eff}(r)$ represent?

The effective potential energy that includes angular momentum contributions. (B) Signup and view all the answers

Which equation represents the effective force defined in the context of effective potential energy?

$F_{eff} = -mrac{dV_{eff}}{dr}$ (C) Signup and view all the answers

When analyzing the extrema of effective potential energy $V_{eff}(r)$, what condition must be satisfied?

$rac{dV_{eff}(r)}{dr} = 0$ (B) Signup and view all the answers

Which expression describes the kinetic energy split into two components in polar coordinates?

$T = rac{1}{2}m(rac{dr}{dt}^2) + rac{L^2}{2mr^2}$ (A) Signup and view all the answers

What does the term $T_ heta = rac{L^2}{2mr^2}$ imply regarding angular momentum and kinetic energy?

Angular momentum affects the angular component of kinetic energy inversely with radius. (A) Signup and view all the answers

What does the effective potential energy $V_{eff}(r) = rac{L^2}{2mr^2} + V(r)$ indicate when analyzing system stability?

It indicates points of stable and unstable equilibrium based on its shape. (C) Signup and view all the answers

From the given potential energy function $V(x) = x^2(x-2)^2$, what derivative must be solved to find the extrema?

$rac{dV}{dx} = 0$ (B) Signup and view all the answers

Flashcards

Energy Conservation in 1D Potential

The total energy of a system in one-dimensional motion under the influence of a potential energy V(x) is conserved, meaning it remains constant over time. This conservation is a consequence of the absence of non-conservative forces.

Total Energy in 1D Motion

The total energy of the system is the sum of the kinetic energy (1/2 * mv^2) and the potential energy V(x).

Turning Points in 1D Potential

Points where the particle changes its direction of motion. They correspond to locations where the kinetic energy is zero and all the energy is potential.

Local Minima in 1D Potential

In a one-dimensional potential, local minima are points where the potential energy is lower than the surrounding values. This is where the particle is likely to be found as it has the lowest potential energy.