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.

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.

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:

• [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.

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}[/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}}[/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.

Interactive Exploration: Energy Skate Park

Energy is constantly changing from one form to another, but the total energy of an isolated system remains constant. In this simulation, you'll investigate the principle of conservation of mechanical energy by observing a skateboarder moving along tracks of different shapes. As the skater moves, gravitational potential energy is converted into kinetic energy and back again. When friction is introduced, some of the mechanical energy is transformed into thermal energy. Experiment by building your own tracks, changing the amount of friction, and selecting different environments such as Earth, the Moon, Jupiter, or outer space. Watch the energy bar charts and graphs as the skater moves, and observe how energy is continuously transferred between different forms while the total energy is conserved.