6.5.  The Conservation of Mechanical Energy

CONCEPTS AT A GLANCE The concept of work and the work-energy theorem have led us to the conclusion that an object can possess two kinds of energy: kinetic energy, KE, and gravitational potential energy, PE. The sum of these two energies is called the total mechanical energy E, so that The Concepts-at-a-Glance chart in Figure 6.15 illustrates this summation. The concept of total mechanical energy will be extremely useful in describing the motion of objects in this and other chapters. Later on, in a number of places, we will update this chart to include other forms of energy.

Figure 6.15  CONCEPTS AT A GLANCE The total mechanical energy E is formed by combining the concepts of kinetic energy and gravitational potential energy. This soccer ball has both types of energy as it sails through the air. (©Nathan Bilow/Allsport/Getty Images)

By rearranging the terms on the right side of Equation 6.7a, the work-energy theorem can be expressed in terms of the total mechanical energy:

(6.7a)

or

(6.8)

Remember: Equation 6.8 is just another form of the work-energy theorem. It states that W_nc, the net work done by external nonconservative forces, changes the total mechanical energy from an initial value of E₀ to a final value of E_f.

The conciseness of the work-energy theorem in the form allows an important basic principle of physics to stand out. This principle is known as the conservation of mechanical energy. Suppose that the net work W_nc done by external nonconservative forces is zero, so Then, Equation 6.8 reduces to

(6.9a)

(6.9b)

Equation 6.9a indicates that the final mechanical energy is equal to the initial mechanical energy. Consequently, the total mechanical energy remains constant all along the path between the initial and final points, never varying from the initial value of E₀. A quantity that remains constant throughout the motion is said to be “conserved.” The fact that the total mechanical energy is conserved when is called the principle of conservation of mechanical energy.

CONCEPTS AT A GLANCE The Concepts-at-a-Glance chart in Figure 6.16 outlines how the conservation of mechanical energy arises naturally from concepts that we have encountered earlier. You might recognize that this chart is an expanded version of the one in Figure 6.5. In the middle are the three concepts that we used to develop the work-energy theorem: Newton’s second law, work, and the equations of kinematics. On the right side of the chart is the work-energy theorem, stated in the form If, as the chart shows, then the final and initial total mechanical energies are equal. In other words, which is the mathematical statement of the conservation of mechanical energy.

Figure 6.16  CONCEPTS AT A GLANCE The work-energy theorem leads to the principle of conservation of mechanical energy under circumstances in which Wnc=0 J where Wnc is the net work done by external nonconservative forces. To the extent that air resistance, a nonconservative force, can be ignored, the total mechanical energy of a skydiver is conserved as he or she falls toward the earth. (©Peter Mason/The Image Bank/Getty Images)

The principle of conservation of mechanical energy offers keen insight into the way in which the physical universe operates. While the sum of the kinetic and potential energies at any point is conserved, the two forms may be interconverted or transformed into one another. Kinetic energy of motion is converted into potential energy of position, for instance, when a moving object coasts up a hill. Conversely, potential energy is converted into kinetic energy when an object is allowed to fall. Figure 6.17 illustrates such transformations of energy for a bobsled run, assuming that nonconservative forces, such as friction and wind resistance, can be ignored. The normal force, being directed perpendicular to the path, does no work. Only the force of gravity does work, so the total mechanical energy E remains constant at all points along the run. The conservation principle is well known for the ease with which it can be applied, as in the following examples.

Figure 6.17  If friction and wind resistance are ignored, a bobsled run illustrates how kinetic and potential energy can be interconverted, while the total mechanical energy remains constant. The total mechanical energy is 600 000 J, being all potential energy at the top and all kinetic energy at the bottom.

A motorcyclist is trying to leap across the canyon shown in Figure 6.18 by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side. Figure 6.18  A daredevil jumping a canyon. Reasoning  Once the cycle leaves the cliff, no forces other than gravity act on the cycle, since air resistance is being ignored. Thus, the work done by external nonconservative forces is zero, Accordingly, the principle of conservation of mechanical energy holds, so the total mechanical energy is the same at the final and initial positions of the motorcycle. We will use this important observation to determine the final speed of the cyclist. Problem solving insight Be on the alert for factors, such as the mass m here in Example 8, that sometimes can be eliminated algebraically when using the conservation of mechanical energy. Solution The principle of conservation of mechanical energy is written as  (6.9b)  The mass m of the rider and cycle can be eliminated algebraically from this equation, since m appears as a factor in every term. Solving for vf gives

Examples 9 and 10 emphasize that the principle of conservation of mechanical energy can be applied even when forces act perpendicular to the path of a moving object.

Some of the following situations are consistent with the principle of conservation of mechanical energy, and some are not. Which ones are consistent with the principle? (a) An object moves uphill with an increasing speed. (b) An object moves uphill with a decreasing speed. (c) An object moves uphill with a constant speed. (d) An object moves downhill with an increasing speed. (e) An object moves downhill with a decreasing speed. (f) An object moves downhill with a constant speed. Background: Kinetic energy, gravitational potential energy, and the principle of conservation of mechanical energy play roles in this question. Consider what the conservation principle implies about the way in which kinetic and potential energies change. For similar questions (including calculational counterparts), consult Self-Assessment Test 6.2. This test is described at the end of this section.

A rope is tied to a tree limb and used by a swimmer to swing into the water below. The person starts from rest with the rope held in the horizontal position, as in Figure 6.19, swings downward, and then lets go of the rope. Three forces act on him: his weight, the tension in the rope, and the force due to air resistance. His initial height h0 and final height hf are known. Considering the nature of these forces, conservative versus nonconservative, can we use the principle of conservation of mechanical energy to find his speed vf at the point where he lets go of the rope? Figure 6.19  During the downward swing, the tension T in the rope acts perpendicular to the circular arc and, hence, does no work on the person. Problem solving insight When nonconservative forces are perpendicular to the motion, we can still use the principle of conservation of mechanical energy, because such ”perpendicular“ forces do no work. Reasoning and Solution The principle of conservation of mechanical energy can be used only if the net work Wnc done by nonconservative forces is zero, The tension and the force due to air resistance are nonconservative forces (see Table 6.2), so we need to inquire whether the net work done by these forces is zero. The tension T is always perpendicular to the circular path of the motion, as shown in the drawing. Thus, the angle q  between the tension and the displacement is 90°. According to Equation 6.1, the work depends on the cosine of this angle, or , which is zero. Thus, the work done by the tension is zero. On the other hand, the force due to air resistance is directed opposite to the motion of the swinging person. The angle q  between this force and the displacement is 180°. The work done by the air resistance is not zero, because the cosine of 180° is not zero. Since the net work done by the two nonconservative forces is not zero, we cannot use the principle of conservation of mechanical energy to determine the final speed. On the other hand, if the force due to air resistance is very small, then the work done by this force is negligible. In this case, the net work done by the nonconservative forces is effectively zero, and the principle of conservation of mechanical energy can be used to determine the final speed of the swimmer. Related Homework: Problem 43

Interactive LearningWare 6.3 A person is standing on the edge of a cliff that is 8.10 m above a lake. Using a rope tied to a tree limb, he swings downward, lets go of the rope, and subsequently splashes into the water. He can let go of the rope either as it is swinging downward or a little later as it is swinging upward. Ignore air resistance and, for each case, determine the speed of the person just before entering the water. Related Homework: Problem 32

The next example illustrates how the conservation of mechanical energy is applied to the breathtaking of a roller coaster.

The tallest and fastest roller coaster in the world is now the Steel Dragon in Mie, Japan (Figure 6.20). The ride includes a vertical drop of 93.5 m. The coaster has a speed of 3.0 m/s at the top of the drop. Neglect friction and find the speed of the riders at the bottom. Figure 6.20  The Steel Dragon roller coaster in Mie, Japan, is a giant. It includes a vertical drop of 93.5 m. (©AFP/Corbis Images) Reasoning  Since we are neglecting friction, we may set the work done by the frictional force equal to zero. A normal force from the seat acts on each rider, but this force is perpendicular to the motion, so it does not do any work. Thus, the work done by external nonconservative forces is zero, and we may use the principle of conservation of mechanical energy to find the speed of the riders at the bottom. Solution The principle of conservation of mechanical energy states that  (6.9b)  The mass m of the rider appears as a factor in every term in this equation and can be eliminated algebraically. Solving for the final speed gives where the vertical drop is h0–hf=93.5 m.

When applying the principle of conservation of mechanical energy to solving problems, we have been using the following reasoning strategy:

Applying the Principle of Conservation of Mechanical Energy    1. Identify the external conservative and nonconservative forces that act on the object. For this principle to apply, the total work done by nonconservative forces must be zero, Wnc=0 J. A nonconservative force that is perpendicular to the displacement of the object does no work, for example.    2. Choose the location where the gravitational potential energy is taken to be zero. This location is arbitrary but must not be changed during the course of solving a problem.    3. Set the final total mechanical energy of the object equal to the initial total mechanical energy, as in Equations 6.9a and 6.9b. The total mechanical energy is the sum of the kinetic and potential energies.

Test your understanding of the material in Sections 6.3, 6.4 and 6.5: ·Gravitational Potential Energy  ·Conservative and Nonconservative Forces  ·Conservation of Mechanical Energy

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