91Ó°ÊÓ

A single conservative force \(F(x)\) acts on a \(1.0 \mathrm{~kg}\) particle that moves along an \(x\) axis. The potential energy \(U(x)\) associated with \(F(x)\) is given by $$U(x)=-4 x e^{-x / 4} \mathrm{~J}$$ where \(x\) is in meters. At \(x=5.0 \mathrm{~m}\) the particle has a kinetic energy of \(2.0 \mathrm{~J} .\) (a) What is the mechanical energy of the system? (b) Make a plot of \(U(x)\) as a function of \(x\) for \(0 \leq x \leq 10 \mathrm{~m}\), and on the same graph draw the line that represents the mechanical energy of the system. Use part (b) to determine (c) the least value of \(x\) the particle can reach and (d) the greatest value of \(x\) the particle can reach. Use part (b) to determine (e) the maximum kinetic energy of the particle and (f) the value of \(x\) at which it occurs. (g) Determine an expression in newtons and meters for \(F(x)\) as a function of \(x .(\mathrm{h})\) For what (finite) value of \(x\) does \(F(x)=0\) ?

Step by step solution

01

Identify the Known Variables

We are given that the potential energy function is \(U(x) = -4x e^{-x/4} \; \text{J}\), the mass of the particle is \(1.0 \; \text{kg}\), and at \(x = 5.0 \; \text{m}\) the kinetic energy \(K\) is \(2.0 \; \text{J}\).

02

Calculate Mechanical Energy

Mechanical energy \(E_{mech}\) is the sum of kinetic energy and potential energy: \[ E_{mech} = K + U(x) \] at \(x = 5.0 \; \text{m}\):\[ E_{mech} = 2.0 \; \text{J} + U(5.0) \]So, calculate \(U(5.0)\):\[ U(5.0) = -4(5)e^{-5/4} \; = -20e^{-1.25} \]approximately \(-5.85 \; \text{J}\).Thus,\[ E_{mech} = 2.0 - 5.85 = -3.85 \; \text{J} \].

03

Plot Potential Energy vs. x

This step requires using a tool such as a graphing calculator or software to plot the function \(U(x) = -4x e^{-x/4}\) over the range \(0 \leq x \leq 10 \; \text{m}\). On the same plot, draw a horizontal line at \(E_{mech} = -3.85 \; \text{J}\) to represent the mechanical energy.

04

Find x-positions from Plot

From the plot, the least value of \(x\) is found where the line \(E_{mech} = -3.85 \; \text{J}\) intersects the potential energy curve first. The greatest value of \(x\) is found at its second intersection with the curve.

05

Calculate Maximum Kinetic Energy

Maximum kinetic energy occurs when potential energy is at a minimum. From the plot, find \(x\) where \(U(x)\) is smallest. Then calculate maximum kinetic energy:\[ K_{max} = E_{mech} - U_{min} \].

06

Find x for Maximum Kinetic Energy

From the plot, the value of \(x\) where kinetic energy is maximum corresponds to the minimum of the potential energy \(U(x)\).

07

Derive Force Expression

The force \(F(x)\) is the negative derivative of the potential energy \(U(x)\):\[ F(x) = -\frac{dU}{dx} \]Calculating,\[ \frac{dU}{dx} = -4 e^{-x/4} + x e^{-x/4} \], so\[ F(x) = -(-4 e^{-x/4} + x e^{-x/4}) = (4 - x)e^{-x/4} \] N.

08

Determine x where F(x) = 0

Set the expression for force to zero:\[ (4 - x)e^{-x/4} = 0 \]We find \(4 - x = 0\) implies \(x = 4\; \text{m}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy

Potential energy is a type of energy associated with the position of an object within a conservative force field. It is stored energy that has the potential to do work. In this exercise, the potential energy associated with the force \(F(x)\) is given by the function \(U(x) = -4x e^{-x / 4}\) J. This expression combines both linear and exponential factors, showing how potential energy can vary with position \(x\) along the axis.

To understand how potential energy changes, consider creating a potential energy curve. This graphical analysis shows how \(U(x)\) behaves over a range of \(x\) values, such as from 0 to 10 meters. By observing this curve, you can determine how potential energy decreases or increases, offering insights into how it impacts the total mechanical energy of the system.

Mechanical Energy

Mechanical energy is the sum of potential energy \(U\) and kinetic energy \(K\) in a system. It represents the total energy available to do work. For this problem, at \(x = 5.0 \; \text{m}\), the mechanical energy \(E_\text{mech}\) is calculated using the formula:

\[ E_\text{mech} = K + U(x) \]

With a given kinetic energy of 2.0 J and the computed potential energy \(U(5.0) \approx -5.85 \; \text{J}\), the mechanical energy is:

\[ E_\text{mech} = 2.0 - 5.85 = -3.85 \; \text{J} \]

Mechanical energy is crucial because it remains constant in a closed system without non-conservative forces like friction. Understanding mechanical energy allows students to predict how energy distributes between kinetic and potential forms as an object moves.

Kinetic Energy

Kinetic energy is the energy of motion. It depends on an object's mass and its velocity. In this context, kinetic energy is given at \(x = 5.0 \; \text{m}\) as 2.0 J. When the particle moves, mechanical energy conversion between kinetic and potential energies occurs.

The maximum kinetic energy occurs when potential energy is at its minimum. By analysing the potential energy graph, you can identify \(x\) positions where \(U(x)\) is minimized and then find the maximum kinetic energy using:

\[ K_{\text{max}} = E_{\text{mech}} - U_{\text{min}} \]

It helps understand the dynamic nature of energy in a system and how kinetic energy maximizes when potential energy reduces.

Derivatives in Physics

Derivatives in physics are used to determine rates of change. They are essential in expressing how quantities vary, such as finding the force from potential energy. The negative derivative of potential energy \(U(x)\) with respect to \(x\) gives the force \(F(x)\):

\[ F(x) = -\frac{dU}{dx} \]

From the calculation in this exercise, \(\frac{dU}{dx}\) is:

\[ \frac{dU}{dx} = -4 e^{-x/4} + x e^{-x/4} \]

Thus, the force is:

\[ F(x) = (4 - x) e^{-x/4} \; \text{N} \]

Using derivatives allows precise quantification of how the object is influenced by forces at different positions, explaining how energy transformations occur in conservative fields.

Graphical Analysis in Physics

Graphical analysis in physics provides a visual representation of how quantities vary. For this problem, plotting the potential energy \(U(x)\) across a range of \(x\) values helps to visualize its behavior over space.

By sketching \(U(x) = -4x e^{-x/4}\) from 0 to 10 meters, and adding a horizontal line representing \(E_{\text{mech}} = -3.85 \; \text{J}\), you can detect key transition points where energy balance changes. Points of intersection signify where potential energy matches mechanical energy, revealing least and greatest \(x\) values the particle can reach.

Graphs are powerful tools, allowing a deeper understanding of energy interplay without tedious calculations, and enabling predictions about system behavior based on visual data. They simplify complex relationships by conveying information efficiently.