Work, Energy, and Energy Resources
45 Conservative Forces and Potential Energy
Learning Objectives
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Define conservative force, potential energy, and mechanical energy.
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Explain the potential energy stored in a spring when Hooke’s law applies.
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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:
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A mass held at a height above the ground (gravitational potential energy).
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A compressed or stretched spring (elastic potential energy).
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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]
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[latex]F[/latex] is the applied force,
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[latex]k[/latex] is the spring constant (stiffness),
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[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.

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.

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:
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[latex]\text{KE}[/latex] is kinetic energy
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[latex]\text{PE}[/latex] is potential energy
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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.

Strategy
The spring force and the gravitational force are conservative forces, so conservation of mechanical energy can be used. Thus,
or
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
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
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
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
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:
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Conservation of mechanical energy in systems with and without friction
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Interplay between KE and PE during motion along curved paths
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Influence of gravitational acceleration on motion and energy
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Real-time graphical visualization of energy transformations
Section Summary
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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.
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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.
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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.
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Mechanical energy is defined as the sum of kinetic and potential energy:
[latex]\text{Mechanical Energy} = \text{KE} + \text{PE}[/latex]
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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
- What is a conservative force?
- 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.
- Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act?
- What is the relationship of potential energy to conservative force?
Problems & Exercises
- 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?
- 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
a force that does the same work for any given initial and final configuration, regardless of the path followed
energy due to position, shape, or configuration
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
the rule that the sum of the kinetic energies and potential energies remains constant if only conservative forces act on and within a system
the sum of kinetic energy and potential energy