A conservative force F(x) acts on a 2.0 kg particle that moves along an x axis. The potential e

step 1 Well, most of the particle M is equal to 2 kilograms. Most of the particle, M, is equal to 2 kil

A conservative force F(x) acts on a 2.0 kg particle that...

A conservative force $F(x)$ acts on a $2.0 \mathrm{~kg}$ particle that moves along an $x$ axis. The potential energy $U(x)$ associated with $F(x)$ is graphed in Fig. $8-63 .$ When the particle is at $x=2.0 \mathrm{~m}$, its velocity is $-1.5 \mathrm{~m} / \mathrm{s}$. What are the (a) magnitude and (b) direction of $F(x)$ at this position? Between what positions on the (c) left and (d) right does the particle move? (e) What is the particle's speed at $x=7.0 \mathrm{~m} ?$

Key Concepts

Conservation of Mechanical Energy

In systems where only conservative forces do work, the total mechanical energy, which is the sum of kinetic and potential energies, remains constant. This principle allows us to relate velocities and potential energies at different positions along the path of a moving particle.

Turning Points in Particle Motion

Turning points occur where the kinetic energy of the particle drops to zero and it changes the direction of motion. These points can be found by equating the total mechanical energy to the potential energy, thereby determining the positions where the particle reverses its course along its motion.

Conservative Forces

Conservative forces are those forces for which the work done in moving a particle between any two points is independent of the path taken. This means the work done can be completely described by changes in the potential energy associated with the force, and thus energy conservation principles are applicable.

Potential Energy and Its Relation to Force

The potential energy function U(x) is directly related to the force acting on a particle by the negative gradient, expressed mathematically as F(x) = -dU/dx. This relation indicates that the force acting on the particle points in the direction in which the potential energy decreases most rapidly.

Transcript

00:01 Well, most of the particle m is equal to 2 kilograms.

00:06 Most of the particle, m, is equal to 2 kilograms.

00:12 And the particle is moving along x -axis.

00:17 Velocity v is equal to 1 .5, it's minus 1 .5 meter per second.

00:26 It x is equal to it x is equal to two meters okay part a well the potential energy of the particle is equal to the negative of work done therefore we can write you potential energy which is a function of x is equal to negative of work done it x is equal to two meters so this is eight x is equal to two meters and since ux is minus eight which is equal to minus w minus w and here eight is equal to and work done is uh the dot product of force and uh displacement and hence we have uh eight is equal to force times displacement which is 2 i cap and therefore force f of x force f is equal to i cap nymphon the magnitude of the conservative force at x is equal to two meters is 4 .25 newton's okay now and this is x is equal to a tax is equal to two meters all right now let's solve part b well the answer to the part b is directed along positive x -axis is directed along positive axis positive x -axis okay now let's solve part c well in part c according to conservation of energy we have e total is equal to some of kinetic energy and potential energy so each total is equal to kinetic energies and potential energy and it's equal to half mv square plus u and putting the values we have one divided by two multiply by two multiply by minus 1 .5 square minus 8 and it's equal to minus 5 .7 jules so e total is equal to minus 5 .75 joules and 8 potential energy at x is equal to minus 5 .75 jules and x is equal to at x is equal to 1 .6 and x is equal to 13 .5 meters 13 .5 meters.

04:33 Well then the particle moves from the left it x is equal to it x is equal to 1 .6 meters.

04:43 It x is equal to 1 .6 meters and now we have part b, part d sorry...

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