Nonconservative Forces in Real Life

In biomechanics and health-related systems, forces are not always conservative. A nonconservative force is one where the work done depends on the specific path taken. The most familiar example is friction, which is ever-present in human motion, medical equipment, and the environment.

Consider rubbing your hands together: the longer the path (more rubbing), the more heat (thermal energy) you generate. This is in contrast to conservative forces, like gravity or spring forces, where only the starting and ending points matter.

Because nonconservative forces dissipate energy (often as heat or sound), they do not have an associated potential energy. Their main effect is to change the total mechanical energy of a system.

Friction as a Nonconservative Force

Figure 46.1 shows that the work done erasing the happy face depends on the path the eraser takes. In (a), the eraser moves directly from A to B. In (b), the longer path requires more work and erases more of the face. Friction removes mechanical energy by converting it to thermal energy, which cannot be fully recovered.

Friction is common in the human body. For example:

• The sliding of tendons over bone surfaces.

• The resistance of blood flow in vessels.

• The drag forces encountered during motion in fluid environments.

In these systems, friction removes mechanical energy from the system, often converting it into heat.

How Nonconservative Forces Affect Mechanical Energy

When nonconservative forces are present, mechanical energy is no longer conserved. Suppose you slide a heavy box across the floor. The force of friction slows it down and generates heat, reducing the system’s total mechanical energy.

For example, when a car is brought to a stop by friction on level ground, it loses kinetic energy, which is dissipated as thermal energy, reducing its mechanical energy. Figure 46.2 compares the effects of conservative and nonconservative forces. We often choose to understand simpler systems such as that described in Figure 46.2(a) first before studying more complicated systems as in Figure 46.2(b).

The Work-Energy Theorem Revisited

We can now update the work-energy theorem to account for both conservative and nonconservative forces. Recall:

[latex]{W}_{\text{net}} = \Delta \text{KE}[/latex]

If both types of forces act:

[latex]{W}{\text{net}} = {W}{\text{c}} + {W}_{\text{nc}}[/latex]

Substituting into the theorem:

[latex]{W}{\text{c}} + {W}{\text{nc}} = \Delta \text{KE}[/latex]

Since conservative work relates to changes in potential energy:

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

Then:

[latex]{W}_{\text{nc}} = \Delta \text{KE} + \Delta \text{PE}[/latex]

This key result shows that the change in mechanical energy (kinetic + potential) is caused by the work done by nonconservative forces.

Applying This to a Ramp

In Figure 46.3 a person pushes a crate up a ramp while friction and gravity resist the motion. The person’s applied force does more work than is lost to friction, so the crate gains mechanical energy. If friction were greater, more energy would be lost as heat.

The final mechanical energy of the system can be calculated using:

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

This equation is extremely useful in physiology and rehabilitation science. For example:

• A physical therapist pushing a patient in a wheelchair uphill must overcome both gravity and friction.

• When a patient walks with a prosthetic leg, some energy is lost to internal friction in the joint components.

Summary of the Effects

• If [latex]{W}_{\text{nc}} > 0[/latex], mechanical energy increases (external work is added).

• If [latex]{W}_{\text{nc}} 0[/latex], mechanical energy decreases (energy is dissipated).

• If [latex]{W}_{\text{nc}} = 0[/latex], mechanical energy is conserved (only conservative forces act).

Key Equation: Mechanical Energy with Nonconservative Forces

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

This general energy equation applies whether you are analyzing a limb in motion, a heart valve moving blood, or a person walking uphill.

Would you like a worked example next, perhaps involving a biological or medical system?

Example 46.1: Calculating Distance Traveled: How Far a Baseball Player Slides

Consider the situation shown in Figure 46.4, where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.

Strategy

Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the work-energy theorem, the work done by friction, which is negative, is added to the initial kinetic energy to reduce it to zero. The work done by friction is negative, because [latex]\mathbf{\text{f}}[/latex] is in the opposite direction of the motion (that is, [latex]\theta =\text{180º}[/latex], and so [latex]\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =-1[/latex]). Thus [latex]{W}_{\text{nc}}=-\text{fd}[/latex]. The equation simplifies to

[latex]\frac{1}{2}{{\text{mv}}_{i}}^{2}-\text{fd}=0[/latex]

or

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

This equation can now be solved for the distance [latex]d[/latex].

Solution

Solving the previous equation for [latex]d[/latex] and substituting known values yields

[latex]\begin{array}{lll}d& =& \frac{{{\text{mv}}_{i}}^{2}}{2f}\\ & =& \frac{(\text{65.0 kg})(6\text{.}\text{00 m/s}{)}^{2}}{(2)(\text{450 N})}\\ & =& \text{2.60 m.}\end{array}[/latex]

Discussion

The most important point of this example is that the amount of nonconservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large mechanical energy, than to stop a mosquito.

Example 46.2: Calculating Distance Traveled: Sliding Up an Incline

Suppose that the player from the previous example is running up a hill having a [latex]5\text{.}\text{00º}[/latex] incline upward with a surface similar to that in the baseball stadium. The player slides with the same initial speed. Determine how far he slides.

 

Strategy

In this case, the work done by the nonconservative friction force on the player reduces the mechanical energy he has from his kinetic energy at zero height, to the final mechanical energy he has by moving through distance [latex]d[/latex] to reach height [latex]h[/latex] along the hill, with [latex]h=d\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}5.00º[/latex]. This is expressed by the equation

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

Solution

The work done by friction is again [latex]{W}_{\text{nc}}=-\text{fd}[/latex]; initially the potential energy is [latex]{\text{PE}}_{i}=\text{mg}\cdot 0=0[/latex] and the kinetic energy is
[latex]{\text{KE}}_{i}=\frac{1}{2}{{\text{mv}}_{i}}^{2}[/latex]; the final energy contributions are
[latex]{\text{KE}}_{f}=0[/latex] for the kinetic energy and [latex]{\text{PE}}_{f}=\text{mgh}=\text{mgd}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta[/latex] for the potential energy.

Substituting these values gives

[latex]\frac{1}{2}{{\text{mv}}_{i}}^{2}+0+\left(-\text{fd}\right)=0+\text{mgd}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\theta .}[/latex]

Solve this for [latex]d[/latex] to obtain

[latex]\begin{array}{lll}d& =& \frac{\left(\frac{1}{2}\right){{\text{mv}}_{\text{i}}}^{2}}{f+\text{mg}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta }\\ & =& \frac{\text{(0.5)}\left(\text{65.0 kg}\right)\left(\text{6.00 m/s}\right)^{2}}{\text{450 N}+\left(\text{65.0 kg}\right)\left({\text{9.80 m/s}}^{2}\right)\phantom{\rule{0.25em}{0ex}}{\text{sin (5.00º)}}^{}}\\ & =& \text{2.31 m.}\end{array}[/latex]

Discussion

As might have been expected, the player slides a shorter distance by sliding uphill. Note that the problem could also have been solved in terms of the forces directly and the work energy theorem, instead of using the potential energy. This method would have required combining the normal force and force of gravity vectors, which no longer cancel each other because they point in different directions, and friction, to find the net force. You could then use the net force and the net work to find the distance [latex]d[/latex] that reduces the kinetic energy to zero. By applying conservation of energy and using the potential energy instead, we need only consider the gravitational potential energy [latex]\text{mgh}[/latex], without combining and resolving force vectors. This simplifies the solution considerably.

Making Connections: Take-Home Investigation—Determining Friction from the Stopping Distance

This hands-on experiment helps illustrate how gravitational potential energy can be converted into thermal energy through the force of friction, an important concept in biomechanics, rehabilitation, and physical therapy.

Materials:

• A ruler with a groove (or a ramp)

• A book (to incline the ruler)

• A marble

• A foam cup with a small hole near the base (see Figure 46.6)

• A steel marble or ball bearing (optional, for comparison)

Procedure:

1. Position the ruler at an incline using the book like in Figure 46.6.

2. From the 10-cm mark, release the marble so it rolls into the foam cup.

3. Measure how far the cup slides across the surface before coming to rest—this distance is [latex]d[/latex].

4. Repeat the trial from the 20-cm and 30-cm positions and record [latex]d[/latex] for each case.

5. Plot distance moved by the cup versus initial height (position) of the marble on the ramp.

   • Is the plot linear?

   • What does that say about energy conversion?