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Work, Energy, and Energy Resources

45 Conservative Forces and Potential Energy

Learning Objectives

  • Define conservative force, potential energy, and mechanical energy.

  • Explain the potential energy stored in a spring when Hooke’s law applies.

  • Use the work-energy theorem to show how conservative forces result in conservation of mechanical energy.

What is a Conservative Force?

In biomechanics and physics, forces do work when they cause movement. Some forces—like gravity and spring forces—have a special property: the work they do depends only on the starting and ending positions, not the path taken. These are called conservative forces.

For example, when you compress a spring in a rehabilitation device or raise a weight during physical therapy, you’re doing work against a conservative force. That energy is stored as potential energy and can be recovered later.

Potential Energy: Stored Energy from Position or Shape

Potential energy (PE) is stored energy that depends on the position, shape, or configuration of an object or system. Some key examples include:

  • A mass held at a height above the ground (gravitational potential energy).

  • A compressed or stretched spring (elastic potential energy).

  • A bent tendon or stretched muscle fiber (elastic deformation energy in biological tissues).

A conservative force always results in a change in potential energy. The energy is recoverable and not lost to heat, unlike with friction.

Spring Potential Energy and Hooke’s Law

Let’s now examine how energy is stored in a spring—something that’s also relevant for tendons and ligaments in the body, which can behave like elastic springs.

A spring follows Hooke’s Law (See Figure 45.1), which says the force needed to stretch or compress it is proportional to the displacement:

[latex]F = kx[/latex]

  • [latex]F[/latex] is the applied force,

  • [latex]k[/latex] is the spring constant (stiffness),

  • [latex]x[/latex] is the displacement from equilibrium.

The potential energy stored in a spring is:

[latex]\text{PE}_{s} = \frac{1}{2}kx^2[/latex]

This means the energy stored grows with both stiffness and the square of the displacement. The same principle applies to a tendon stretching under load.

An undeformed spring fixed at one end with no potential energy. (b) A spring fixed at one end and stretched by a distance x by a force F equal to k x. Work done W is equal to one half k x squared. P E s is equal to one half k x squared. (c) A graph of force F versus elongation x in the spring. A straight line inclined to x axis starts from origin. The area under this line forms a right triangle with base of x and height of k x. Area of this triangle is equal to one half k x squared.
Figure 45.1: (a) An undeformed spring has no [latex]{\text{PE}}_{s}[/latex] stored in it. (b) The force needed to stretch (or compress) the spring a distance [latex]x[/latex] has a magnitude [latex]F=\text{kx}[/latex] , and the work done to stretch (or compress) it is [latex]\frac{1}{2}{\text{kx}}^{2}[/latex]. Because the force is conservative, this work is stored as potential energy [latex]\left({\text{PE}}_{s}\right)[/latex] in the spring, and it can be fully recovered. (c) A graph of [latex]F[/latex] vs. [latex]x[/latex] has a slope of [latex]k[/latex], and the area under the graph is [latex]\frac{1}{2}{\text{kx}}^{2}[/latex]. Thus the work done or potential energy stored is [latex]\frac{1}{2}{\text{kx}}^{2}[/latex].

The equation [latex]{\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}[/latex] has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy is energy due to position, shape, or configuration. For shape or position deformations, stored energy is [latex]{\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}[/latex], where [latex]k[/latex] is the force constant of the particular system and [latex]x[/latex] is its deformation. Another example is seen in Figure 45.2 for a guitar string. Similar energy storage occurs in tendons, ligaments, and even in the arches of the feet, which compress and rebound during walking and running.

A six-string guitar is placed vertically. The left-most string is plucked in the left direction with a force F shown by an arrow pointing left. The displacement of the string from the mean position is d. The plucked string is labeled P E sub string, to represent the potential energy of the string.
Figure 45.2: Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string.

Conservation of Mechanical Energy

When only conservative forces are involved, energy is conserved within the system. The work-energy theorem states:

[latex]W_{\text{net}} = \Delta \text{KE} = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2[/latex]

If only conservative forces act:

[latex]W_{\text{net}} = W_{\text{c}}[/latex]

And because work done by conservative forces equals the negative change in potential energy:

[latex]W_{\text{c}} = -\Delta \text{PE}[/latex]

Combining both gives:

[latex]\Delta \text{KE} + \Delta \text{PE} = 0[/latex]

or

[latex]\text{KE} + \text{PE} = \text{constant}[/latex]

Which leads to the principle of conservation of mechanical energy:

[latex]\text{KE}_i + \text{PE}_i = \text{KE}_f + \text{PE}_f[/latex]

Where:

  • [latex]\text{KE}[/latex] is kinetic energy

  • [latex]\text{PE}[/latex] is potential energy

  • Subscripts [latex]i[/latex] and [latex]f[/latex] represent initial and final states

This principle holds true as long as all forces are conservative—no friction or energy losses.

Example 45.1: Using Conservation of Mechanical Energy to Calculate the Speed of a Toy Car

A 0.100-kg toy car is propelled by a compressed spring, as shown in Figure 45.3. The car follows a track that rises 0.180 m above the starting point. The spring is compressed 4.00 cm and has a force constant of 250.0 N/m. Assuming work done by friction to be negligible, find (a) how fast the car is going before it starts up the slope and (b) how fast it is going at the top of the slope.

The figure shows a toy race car that has just been released from a spring. Two possible paths for the car are shown. One path has a gradual upward incline, leveling off at a height of eighteen centimeters above its starting level. An alternative path shows the car descending from its starting point, making a loop, and then ascending back up and leveling off at a height of eighteen centimeters above its starting level.
Figure 45.3: A toy car is pushed by a compressed spring and coasts up a slope. Assuming negligible friction, the potential energy in the spring is first completely converted to kinetic energy, and then to a combination of kinetic and gravitational potential energy as the car rises. The details of the path are unimportant because all forces are conservative—the car would have the same final speed if it took the alternate path shown.

Strategy

The spring force and the gravitational force are conservative forces, so conservation of mechanical energy can be used. Thus,

[latex]{\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}[/latex]

or

[latex]\frac{1}{2}{{\text{mv}}_{i}}^{2}+{\text{mgh}}_{i}+\frac{1}{2}{{\text{kx}}_{i}}^{2}=\frac{1}{2}{{\text{mv}}_{f}}^{2}+{\text{mgh}}_{f}+\frac{1}{2}{{\text{kx}}_{f}}^{2},[/latex]

where [latex]h[/latex] is the height (vertical position) and [latex]x[/latex] is the compression of the spring. This general statement looks complex but becomes much simpler when we start considering specific situations. First, we must identify the initial and final conditions in a problem; then, we enter them into the last equation to solve for an unknown.

Solution for (a)

This part of the problem is limited to conditions just before the car is released and just after it leaves the spring. Take the initial height to be zero, so that both [latex]{h}_{\text{i}}[/latex] and [latex]{h}_{\text{f}}[/latex] are zero. Furthermore, the initial speed [latex]{v}_{\text{i}}[/latex] is zero and the final compression of the spring [latex]{x}_{\text{f}}[/latex] is zero, and so several terms in the conservation of mechanical energy equation are zero and it simplifies to

[latex]\frac{1}{2}{{\text{kx}}_{i}}^{2}=\frac{1}{2}{{\text{mv}}_{f}}^{2}\text{.}[/latex]

In other words, the initial potential energy in the spring is converted completely to kinetic energy in the absence of friction. Solving for the final speed and entering known values yields

[latex]\begin{array}{lll}{v}_{f}& =& \sqrt{\frac{k}{m}}{x}_{i}\\ & =& \sqrt{\frac{\text{250}\text{.0 N/m}}{\text{0.100 kg}}}\left(\text{0.0400 m}\right)\\ & =& \text{2.00 m/s.}\end{array}[/latex]

Solution for (b)

One method of finding the speed at the top of the slope is to consider conditions just before the car is released and just after it reaches the top of the slope, completely ignoring everything in between. Doing the same type of analysis to find which terms are zero, the conservation of mechanical energy becomes

[latex]\frac{1}{2}{\text{kx}}_{\text{i}}^{ 2}=\frac{1}{ 2}{\text{mv}}_{\text{f}}^{\text{ 2}}+{\text{mgh}}_{\text{f}}\text{.}[/latex]

This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy and partly to kinetic energy. The final speed at the top of the slope will be less than at the bottom. Solving for [latex]{v}_{\text{f}}[/latex] and substituting known values gives

[latex]\begin{array}{lll}{v}_{f}& =& \sqrt{\frac{{{\text{kx}}_{i}}^{2}}{m}-2{\text{gh}}_{f}}\\ & =& \sqrt{\left(\frac{\text{250.0 N/m}}{\text{0.100 kg}}\right)\left(\text{0.0400 m}\right)^{2}-2\left(\text{9.80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)\left(\text{0.180 m}\right)}\\ & =& \text{0.687 m/s.}\end{array}[/latex]

Discussion

Another way to solve this problem is to realize that the car’s kinetic energy before it goes up the slope is converted partly to potential energy—that is, to take the final conditions in part (a) to be the initial conditions in part (b).

Important Note on Conservative Forces

For conservative forces, we do not need to calculate the work done directly. Instead, we analyze their effects through changes in potential energy, as demonstrated in the previous example. A key advantage of conservative forces is that their influence depends only on the initial and final positions of the system—not on the specific path taken. This greatly simplifies problem-solving, especially when the actual path is complex or when the forces vary along the trajectory. As long as the chosen path is physically possible, its details are irrelevant when applying the principle of conservation of mechanical energy.

PhET Explorations: Energy Skate Park

Explore the principle of conservation of mechanical energy in an engaging, interactive environment.

In this simulation, you can build tracks, ramps, and jumps for a skateboarder and observe how energy transforms between kinetic energy (KE), gravitational potential energy (PE), and thermal energy due to friction. Visual graphs and energy bar charts dynamically display these quantities as the skater moves.

You can also place the simulation in various environments—including Earth, the Moon, Jupiter, and even outer space—to explore how gravitational strength affects energy transformations.

Key Concepts Explored:

  • Conservation of mechanical energy in systems with and without friction

  • Interplay between KE and PE during motion along curved paths

  • Influence of gravitational acceleration on motion and energy

  • Real-time graphical visualization of energy transformations

Section Summary

    • A conservative force is one for which the work done depends only on the starting and ending points of a motion—not on the path taken.

    • For every conservative force, we can define a corresponding potential energy [latex]\left(\text{PE}\right)[/latex], just as we defined [latex]{\text{PE}}_{g}[/latex] for gravity.

    • The potential energy stored in a spring is given by:

      [latex]{\text{PE}}_{s} = \frac{1}{2} kx^{2}[/latex]

      where [latex]k[/latex] is the spring constant and [latex]x[/latex] is the displacement from the spring’s equilibrium (undeformed) position.

    • Mechanical energy is defined as the sum of kinetic and potential energy:

      [latex]\text{Mechanical Energy} = \text{KE} + \text{PE}[/latex]

    • When only conservative forces act within a system, the total mechanical energy remains constant. This is expressed by the conservation of mechanical energy:

      [latex]\begin{array}{cc} & \text{KE} + \text{PE} = \text{constant} \ \text{or} & \ & \text{KE}{\text{i}} + \text{PE}{\text{i}} = \text{KE}{\text{f}} + \text{PE}{\text{f}} \end{array}[/latex]

      where [latex]i[/latex] and [latex]f[/latex] denote the initial and final states of the system.

    Conceptual Questions

    1. What is a conservative force?
    2. The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer dives from it, starting just before the swimmer steps on the board until just after his feet leave it.
    3. Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act?
    4. What is the relationship of potential energy to conservative force?

    Problems & Exercises

    1. A [latex]5\text{.}\text{00}×{\text{10}}^{5}\text{-kg}[/latex] subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant [latex]k[/latex] of the spring?
    2. A pogo stick has a spring with a force constant of [latex]2\text{.}\text{50}×{\text{10}}^{4}\phantom{\rule{0.20em}{0ex}}\text{N/m}[/latex], which can be compressed 12.0 cm. To what maximum height can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40.0 kg? Explicitly show how you follow the steps in the Problem-Solving Strategies for Energy.

    Glossary

    conservative force
    a force that does the same work for any given initial and final configuration, regardless of the path followed
    potential energy
    energy due to position, shape, or configuration
    potential energy of a spring
    the stored energy of a spring as a function of its displacement; when Hooke’s law applies, it is given by the expression [latex]\frac{1}{2}{\text{kx}}^{2}[/latex] where [latex]x[/latex] is the distance the spring is compressed or extended and [latex]k[/latex] is the spring constant
    conservation of mechanical energy
    the rule that the sum of the kinetic energies and potential energies remains constant if only conservative forces act on and within a system
    mechanical energy
    the sum of kinetic energy and potential energy
    definition

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