Section 4–3 Kinetic energy

(Pendulum / Kinetic energy / Relativistic correction)

This section on kinetic energy is very brief as compared to potential energy. The three interesting points discussed are a pendulum, kinetic energy, and relativistic correction for the equation of kinetic energy.

1. Pendulum:

“…To illustrate another type of energy we consider a pendulum. If we pull the mass aside and release it, it swings back and forth. In its motion, it loses height in going from either end to the center (Feynman et al., 1963, section 4.3 Kinetic energy).”

Interestingly, Feynman demonstrates the law of conservation of energy by using a pendulum and he is confident that the pendulum would not rise to a greater height to hit him. He initially brings the pendulum to a height as high as his chin and then releases it. While the pendulum is swinging to and fro, it is possible to have a continuous change of gravitational potential energy to kinetic energy, and vice versa. During its motion, it may lose height in going from either end to the center with a gain in kinetic energy and a loss in gravitational potential energy. When the pendulum climbs up towards our chin, there is a loss in kinetic energy and a gain in gravitational potential energy instead.

One may google “Walter Lewin Kinetic & Potential Energy” to have a look at a similar demonstration by using a pendulum. In the video, Lewin explains that “physics works” and he is still alive because the pendulum did not hit him. Strictly speaking, it is possible that the pendulum gains additional kinetic energy during an earthquake.

2. Kinetic energy:

“…The kinetic energy at the bottom equals the weight times the height that it could go, corresponding to its velocity: K.E. = WH. ... We shall soon find that we can write it this way: K.E. = WV²/2g (Feynman et al., 1963, section 4.3 Kinetic energy).”

By using arguments about reversible machines (or law of conservation of energy), we can see that a pendulum at the lowest point of its path must have a quantity of energy which permits it to rise a certain height. Feynman says that this has nothing to do with any machinery by which the pendulum comes up or the path by which it ascends. Importantly, the kinetic energy of the pendulum at the lowest height equals to the weight of the pendulum times the height that it could go (K.E. = W ´ H). Furthermore, the kinetic energy depends on the velocity and a formula for kinetic energy of an object can be derived as K.E. = WV²/2g. (Feynman shows that the formula for kinetic energy K.E. = ½ mV² is correct in chapter 13.)

Note: During a British Broadcasting Corporation interview, Feynman (1994) explains that “[w]hen a ball comes down and bounces, it shakes irregularly some of the atoms in the floor, and when it comes up again, it has left some of those floor atoms moving, jiggling. As it bounces, it is passing its extra energy, extra motions to little patches on the floor (p. 127).” In short, a percentage of the kinetic energy of a macroscopic object is converted into the kinetic energy of microscopic objects (or atoms) during a collision.

3. Relativistic correction:

“…Both (4.3) and (4.6) are approximate formulas, the first because it is incorrect when the heights are great, i.e., when the heights are so high that gravity is weakening; the second, because of the relativistic correction at high speeds (Feynman et al., 1963, section 4.3 Kinetic energy).”

The equation K.E. = WV²/2g is a general formula for kinetic energy of an object moving with velocity V. Curiously, the motion of an object has kinetic energy has nothing to do with the fact that we are in a gravitational field. This is because the gravitational field g can be eliminated by writing W as mg such that the formula is simplified as K.E. = ½ mV². However, the kinetic energy of pendulum can be transformed from the energy stored in the gravitational field (or gravitational potential energy). Moreover, the equation for kinetic energy is only approximately correct because there is a need for relativistic correction when the object is moving at very high speeds. Thus, the law of conservation of energy is strictly true when we apply the exact formula for all forms of energy.

Note: The reality (or objectivity) of kinetic energy was questioned because it can be assigned any value depending on an observer’s frame of reference. For example, Heaviside (1891) has questioned the reality of energy due to the relative motion: “[b]ut we need not go so far as to assume the objectivity of energy. This is an exceedingly difficult notion, and seems to be rendered inadmissible by the mere fact of the relativity of motion, on which kinetic energy depends (p.425).” As an example, a book on a table may appear at rest from the perspective of an observer on earth. However, one may explain that the object is moving at a high speed due to the rotation of the earth. In other words, the kinetic energy of the object is not an invariant.

Questions for discussion:

1. Is it absolutely safe to demonstrate the law of conservation of energy by using some heavy bricks as a pendulum?

2. Does the kinetic energy of an object depend on the gravitational field (K.E. = WV²/2g)?

3. What is the limitation of formula for kinetic energy, K.E. = ½ mV²?

The moral of the lesson: the kinetic energy of an object is dependent on its velocity but the formula needs a relativistic correction when the object is moving at very high speeds (especially approaching the speed of light).

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company.

3. Heaviside, O. (1891). On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field. Proceedings of the Royal Society of London, 50, 126–129.