Define conservative force, potential energy, and mechanical energy.Explain the potential power of a spring in regards to its compression once Hooke’s law applies.Use the work-energy organize to show how having only conservative forces implies preservation of mechanical energy.

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## Potential Energy and Conservative Forces

Work is excellent by a force, and also some forces, such together weight, have actually special characteristics. A conservative force is one, favor the gravitational force, for which work-related done through or versus it depends only on the starting and finishing points that a motion and also not on the path taken. Us can define a potential energy (PE) for any kind of conservative force, simply as we did because that the gravitational force. Because that example, once you wind increase a toy, an egg timer, or an old-fashioned watch, you carry out work against its spring and also store power in it. (We treat these springs together ideal, in that us assume there is no friction and also no production of thermal energy.) This stored energy is recoverable together work, and it is beneficial to think that it together potential energy had in the spring. Indeed, the reason that the spring has this properties is the its force is conservative. The is, a conservative pressure results in stored or potential energy. Gravitational potential energy is one example, as is the power stored in a spring. We will additionally see just how conservative forces are related to the conservation of energy.

### Potential Energy and also Conservative Forces

Potential power is the power a system has due to position, shape, or configuration. It is stored power that is totally recoverable.

A conservative pressure is one for which work done by or against it depends just on the beginning and finishing points the a motion and also not on the course taken.

We can specify a potential power (PE) for any kind of conservative force. The occupational done against a conservative pressure to with a final configuration counts on the configuration, not the course followed, and is the potential power added.

## Potential power of a Spring

First, allow us obtain an expression because that the potential power stored in a feather (PEs). Us calculate the work-related done to stretch or compress a spring the obeys Hooke’s law. (Hooke’s regulation was check in Elasticity: Stress and Strain, and also states that the magnitude of pressure F top top the spring and the result deformation ΔL space proportional, kΔL.) (See number 1.) for our spring, us will replace ΔL (the lot of deformation produced by a force F) by the street x the the feather is stretched or compressed follow me its length. So the pressure needed to stretch the spring has actually magnitude Fkx, whereby k is the spring’s force constant. The pressure increases linearly from 0 in ~ the start to kx in the totally stretched position. The average force is frackx2\. For this reason the work-related done in extending or compressing the spring is W_ exts=Fd=left(frackx2 ight)x=frac12kx^2\. Alternatively, we detailed in Kinetic Energy and also the Work-Energy Theorem that the area under a graph of F vs. x is the work-related done by the force. In number 1c we check out that this area is likewise frac12kx^2\. We as such define the potential power of a spring, PEs, come be

extPE_ exts=frac12kx^2\,

where k is the spring’s force continuous and x is the displacement native its undeformed position. The potential energy represents the occupational done on the spring and also the power stored in it as a an outcome of stretching or compressing it a street x. The potential energy of the feather PEs does not count on the course taken; that depends just on the large or squeeze out x in the final configuration.

Figure 1. (a) an undeformed spring has actually no PEs save on computer in it. (b) The pressure needed to stretch (or compress) the feather a distance x has a size F = kx, and the occupational done to stretch (or compress) the is frac12kx^2\. Due to the fact that the force is conservative, this job-related is stored as potential energy (PEs) in the spring, and also it can be completely recovered. (c) A graph the F vs. X has a steep of k, and the area under the graph is frac12kx^2\. For this reason the work-related done or potential energy stored is frac12kx^2\.

The equation extPE_ exts=frac12kx^2\ has general validity past the special case for i m sorry it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general meaning of potential energy is energy due to position, shape, or configuration. For shape or position deformations, stored energy is extPE_ exts=frac12kx^2\, where k is the force constant of the details system and also x is its deformation. An additional example is checked out in figure 2 for a guitar string.

Figure 2. Work-related is excellent to deform the etc string, providing it potential energy. Once released, the potential energy is converted to kinetic power and back to potential together the string oscillates ago and forth. A an extremely small fraction is dissipated together sound energy, progressively removing energy from the string.

## Conservation of mechanical Energy

Let united state now consider what type the work-energy to organize takes as soon as only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net job-related done through all pressures acting top top a system equates to its readjust in kinetic energy. In equation form, this is W_ extnet=frac12mv^2-frac12mv_0^2=Delta extKE\.

If just conservative forces act, then Wnet = Wc, where Wc is the full work excellent by every conservative forces. Thus, Wc = ΔKE.

Now, if the conservative force, such together the gravitational force or a spring force, does work, the mechanism loses potential energy. That is, Wc=−ΔPE. Therefore, −ΔPE = ΔKE or ΔKE + ΔPE = 0.

This equation method that the total kinetic and potential energy is constant for any procedure involving only conservative forces. That is,

ext(conservative forces only),egincases extKE+ extPE= extconstant\ extor\ extKE_ exti+ extPE_ exti= extKE_ extf+ extPE_ extfendcases\

where i and also f denote initial and also final values. This equation is a type of the work-energy theorem for conservative forces; that is known as the conservation of mechanical energy principle. Remember that this uses to the extent that all the pressures are conservative, so the friction is negligible. The full kinetic add to potential energy of a device is identified to be its mechanically energy, (KE+PE). In a device that experiences just conservative forces, there is a potential energy associated with each force, and also the energy only changes type between KE and the various types of PE, through the total energy remaining constant.

### Example 1. Making use of Conservation of Mechanical energy to calculation the speed of a Toy Car

A 0.100-kg toy car is thrust by a compressed spring, as displayed in figure 3. The car follows a track that rises 0.180 m above the beginning point. The spring is compressed 4.00 cm and also has a force consistent of 250.0 N/m. Assuming job-related done by friction to it is in negligible, uncover the following:

How rapid is the vehicle going before it starts increase the slope?How quick is the going at the top of the slope?

Figure 3. A toy car is moved by a compressed spring and also coasts up a slope. Suspect negligible friction, the potential energy in the feather is first completely convert to kinetic energy, and also then come a mix of kinetic and also gravitational potential power as the automobile rises. The details of the course are unimportant since all pressures are conservative—the car would have actually the same last speed if it take it the alternative path shown.

Strategy

The feather force and the gravitational force are conservative forces, so conservation that mechanical power can it is in used. Thus,

KEi + PEi = KEf + PEf

or

frac12mv_ exti^2+mgh_ exti+frac12kx_ exti^2=frac12mv_ extf^2+mgh_ extf+frac12kx_ extf^2\,

where h is the elevation (vertical position) and x is the compression that the spring. This general statement looks complex but becomes much easier when we start considering specific situations. First, we must determine the initial and also final problems in a problem; then, we go into them right into the critical equation to deal with for an unknown.

Solution for Part 1

This component of the difficulty is limited to conditions just before the vehicle is released and also just after it pipeline the spring. Take the initial elevation to it is in zero, so that both hi and also hf room zero. Furthermore, the initial speed vi is zero and the final compression the the feather xf is zero, and so number of terms in the conservation of mechanical energy equation are zero and also it simplifies to

frac12kx_ exti^2=frac12mv_ extf^2\.

In other words, the early stage potential energy in the feather is converted totally to kinetic energy in the absence of friction. Resolving for the last speed and entering recognized values yields

eginarraylllv_ extf&=&sqrtfrackmx_ exti\ ext &=&sqrtfrac250.0 ext N/m0.100 ext kgleft(0.0400 ext m ight)\ ext &=&2.00 ext m/sendarray\

Solution for Part 2

One method of finding the rate at the height of the steep is come consider conditions just prior to the auto is released and just after it reaches the optimal of the slope, completely ignoring everything in between. Law the same type of evaluation to find which terms room zero, the preservation of mechanical energy becomes

frac12kx_ exti^2=frac12mv_ extf^2+mgh_ extf\.

This type of the equation means that the spring’s early potential energy is converted partly to gravitational potential energy and also partly to kinetic energy. The last speed in ~ the height of the slope will certainly be less than in ~ the bottom. Fixing for vf and also substituting recognized values gives

eginarraylllv_ extf&=&sqrtfrackx_ exti^2m-2gh_ extf\ ext &=&sqrtleft(frac250.0 ext N/m0.100 ext kg ight)left(0.0400 ext m ight)^2-2left(9.80 ext m/s^2 ight)left(0.180 ext m ight)\ ext &=&0.687 ext m/sendarray\

Discussion

Another means to deal with this problem is to realize the the car’s kinetic energy before it goes up the steep is converted partly to potential energy—that is, to take the final conditions in part 1 to it is in the initial conditions in part 2.

Note that, because that conservative forces, we do not directly calculate the occupational they do; rather, we think about their effects through their corresponding potential energies, simply as us did in instance 1. Note likewise that we perform not take into consideration details that the path taken—only the beginning and ending points are necessary (as lengthy as the route is no impossible). This presumption is normally a incredible simplification, since the path might be complex and pressures may vary along the way.

## PhET Explorations: power Skate Park

Learn around conservation of power with a ska dude! develop tracks, ramps and jumps because that the skater and view the kinetic energy, potential energy and friction together he moves. Friend can likewise take the ska to different planets or even space!

## Section Summary

A conservative force is one because that which work depends just on the beginning and ending points the a motion, not on the course taken.We can specify potential energy (PE) for any kind of conservative force, simply as we defined PEg for the gravitational force.The potential energy of a spring is extPE_s=frac12 extkx^2\, where k is the spring’s force consistent and x is the displacement indigenous its undeformed position.Mechanical energy is defined to it is in KE + PE for a conservative force.When just conservative forces act on and also within a system, the complete mechanical energy is constant. In equation form,egincases extKE+ extPE= extconstant\ extor\ extKE_ exti+ extPE_ exti= extKE_ extf+ extPE_ extfendcases\where i and f represent initial and also final values. This is recognized as the preservation of mechanical energy.

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### Conceptual Questions

What is a conservative force?The pressure exerted by a diving plank is conservative, detailed the inner friction is negligible. Assuming friction is negligible, describe transforms in the potential power of a diving board together a swimmer dives from it, starting just prior to the swimmer steps on the plank until simply after his feet leaving it.Define mechanically energy. What is the partnership of mechanical energy to nonconservative forces? What wake up to mechanical power if only conservative forces act?What is the connection of potential power to conservative force?

### Problems & Exercises

A 5.00 × 105-kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a huge spring bumper in ~ the end of that track. What is the force continuous k of the spring?A pogo stick has actually a spring through a force consistent of 2.50 × 104 N/m, which can be compressed 12.0 cm. To what maximum height have the right to a child jump ~ above the pole using only the power in the spring, if the child and also stick have a total mass that 40.0 kg? Explicitly show how you monitor the procedures in the Problem-Solving strategies for Energy.

## Glossary

conservative force: a force that does the same work for any given initial and also final configuration, nevertheless of the course followed

potential energy: energy due to position, shape, or configuration

potential power of a spring: the stored power of a spring together a duty of that is displacement; when Hooke’s legislation applies, it is provided by the expression frac12 extkx^2\ wherein x is the distance the spring is compressed or extended and k is the feather constant

conservation of mechanical energy: the dominion that the amount of the kinetic energies and also potential energies remains continuous if just conservative pressures act on and also within a system