A single conservative force F(x) acts on a 1.0 kg particle that moves along an x axis. The pote

A single conservative force F(x) acts on a 1.0 kg particle...

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 $x$. (h) For what (finite) value of $x$ does $F(x)=0 ?$

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 $x$. (h) For what (finite) value of $x$ does $F(x)=0 ?$

Key Concepts

Conservative Forces

A conservative force is one for which the work done moving a particle between two points is independent of the path taken. This means that the energy associated with the force can be completely described by a scalar potential energy function U(x), and mechanical energy (the sum of kinetic and potential energies) is conserved within the system.

Potential Energy Function

The potential energy function U(x) is a scalar function that describes the energy stored in a system due to the position of a particle in a conservative field. It plays a central role in analyzing the motion of particles, particularly in determining how energy is converted between kinetic and potential forms and in locating points of equilibrium and turning points in the motion.

Mechanical Energy Conservation

Mechanical energy conservation is the principle that the total mechanical energy of a system (the sum of kinetic and potential energies) remains constant if only conservative forces act on it. This concept allows one to relate the particle's velocity and position throughout its motion and to identify its extreme positions where the kinetic energy becomes zero.

Kinetic Energy and Its Relationship to Potential Energy

Kinetic energy is the energy associated with the motion of a particle and, when combined with the potential energy in conservative systems, forms the total mechanical energy. The exchange between kinetic and potential energies as a particle moves under a conservative force is key to understanding the dynamics of the system, with maximum kinetic energy typically occurring at points of minimum potential energy.

Turning Points and Energy Graphs

Turning points in the motion of a particle occur where its kinetic energy becomes zero, meaning all the mechanical energy has been converted to potential energy. Graphical analysis of U(x) versus x, along with the constant mechanical energy line, helps in visually identifying these boundaries and understanding the particle's range of motion within the potential field.

Force as the Negative Gradient of Potential Energy

In conservative systems, the force acting on a particle is given by the negative derivative of the potential energy function with respect to position, F(x) = -dU/dx. This relationship is fundamental in linking the potential energy landscape to the actual forces experienced by the particle and thus to its subsequent motion.

Transcript

00:01 Potential energy is a function of x and potential energy is equal to minus 4 times x times e to the power minus x divided by 4 and this is the potential energy and mass is given which is equal to 1 kilogram.

00:33 Is equal to 1 kilogram.

00:36 We have also k, the kinetic energy, which is equal to 2 joules, and x is equal to 5 meters, 5 meters.

00:56 Now let's solve part a.

00:58 We know that mechanical energy is the sum of potential energy and kinetic energy, therefore e -mechanical.

01:06 E -mechanical is equal to kinetic energy plus potential energy.

01:18 And by substitution, to find the potential energy, we have u 8 .5, at x is equal to 5, is equal to minus 4 times x is 5, minus 4 times 5 times, e to the power, minus 5 divided by 4 and therefore this is equal to minus 5 .7 joules so potential energy at x is equal to 5 is equal to minus 5 .7 joules now e mechanical e mechanical or mechanical energy is equal to kinetic energy plus potential energy and kinetic energy is equal to two joules and potential energy is equal to minus 5 .37 joules therefore we have 2 minus we have 2 minus 5 .7 and it's equal to is equal to is equal to is equal to minus 3 .7 jules minus 3 .7 jules.

03:08 This is the total mechanical energy.

03:13 Now let's do part b.

03:17 In part b we have to plot the graph.

03:20 We have ux potential energy is a function of x and it's equal to minus 4 times x times e to the power minus x divided by 4.

03:38 Now let's plot a graph, a graph between u and x.

03:45 So this is x -axis and this is y -axis.

03:54 And let's take potential energy on the y -axis.

04:01 Ux and this is the x coordinate all right and this is zero and then we have this point is 2 4 and then we have 6 8 10 and so on and similarly here on y x is minus 1 minus 2 minus 4 minus 4 minus 5 6 and the graph looks like this well this is the graph between x and ux now let's go for part c from the graph the least value of x the particle can reach is it x is equal to minus 3 .7 where x is approximately equal to 1 .27 meters so it u is equal to minus 3 .7 it u is equal to this is part c remember it u is equal to minus 3 .7 jules minus 3 .7 jules where x is equal to where x is equal to 1 .27 meters now let's plot the graph.

06:03 This is again the x -axis and we have this y -axis and the graph looks like this and this is x -coordinate and this is potential energy ux in negative values.

06:27 So let's say here we have a point this is minus 3.

06:34 Minus 3 and then here we have minus 4 and this is somewhere 2 and we have this point here this one and which is minus 3 .7 somewhere here minus 3 .7 on the ux and then it is 1 .27 somewhere here so 1 .27 here.

07:13 All right.

07:15 And now let's do part d...

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