Section 19–4 Rotational kinetic energy

(Centrifugal force / Coriolis force / Centripetal force)

In this section, Feynman discusses the concept of centrifugal force, Coriolis force, and centripetal force from a perspective of work done on an object during rotation. His discussion of rotational kinetic energy is uncommon because he did not include typical examples such as an object rolling down a slope that has both translational kinetic energy and rotational kinetic energy. To be more precise, this section could be titled “Coriolis force” because this is his main focus. Instead of ending the chapter abruptly, there could be a mention of real-life examples relating to the Coriolis force such as long-range artillery trajectories, Foucault pendulum, or typhoons.

1. Centrifugal force:

“When we are rotating, there is a centrifugal force on the weights (Feynman et al., 1963, section 19–4 Rotational kinetic energy).”

According to Feynman, there is a centrifugal force on the weights when they are being rotated. An observer in an inertial frame of reference may elaborate that there is actually a tension (centripetal force) that pulls the weights. Feynman adds that the work we do against the centrifugal force ought to agree with the difference in the rotational kinetic energy of the weights. As a suggestion, one may explain that the rotational work should be done by a real force such as frictional force. In short, the centrifugal force exists as a fictitious (pseudo) force and it is a useful concept for an observer in a rotating frame of reference.

Feynman clarifies that the rotational work cannot be contributed by the centrifugal force because it is a radial force. This means that the centrifugal force is not the entire story in a rotating system. In chapter 12, Feynman says that “[a]nother example of pseudo force is what is often called ‘centrifugal force’ (Feynman et al., 1963, section 12–5 Pseudo forces).” However, physics teachers may derive the centrifugal force using (dr/dt)_fixed = (dr/dt)_rotating + w ´ r to help understanding the nature of forces in a rotating frame of reference in which r is a displacement vector (Thornton & Marion, 2004). In the derivation, we can identify three forces: (1) centrifugal force (proportional to w²), (2) Coriolis force (proportional to w), and (3) Euler force (proportional to dw/dt).

2. Coriolis force:

“Now let us develop a formula to show how this Coriolis force really works (Feynman et al., 1963, section 19–4 Rotational kinetic energy).”

Feynman explains that the Coriolis force has a very strange property: an object moving in a rotating system would appear to be pushed sidewise. For example, a person moving radially inward on a carousel would feel a sidewise force that is parallel to −2mω × v. We should clarify that the Coriolis force does arise from the motion of the object because the force is velocity-dependent and vanishes if it stops. In a sense, we may feel the existence of Coriolis force (or effect) when we walk radially inward or outward on a carousel. Importantly, the Coriolis force is a fictitious force that does not obey Newton’s third law (action = reaction) because it does not arise from an interaction between two objects.

Feynman suggests his students walk along the radius of a carousel whereby one has to lean over and push sidewise. (If we cannot counter this sidewise force with a frictional force of magnitude 2mωv, we would experience a tangential acceleration of 2ωv.) Feynman simply derives a formula of Coriolis force using τ = F_cr = dL/dt = d(mωr²)/dt = 2mωr(dr/dt), in which F_c is the Coriolis force. If we do not assume the angular speed to be constant, we can obtain mr²(dω/dt) that is known as the Euler force. However, this derivation does not help to understand the factor of 2 in 2mωv. One may prefer a physical explanation of the two equal components of Coriolis force using a general derivation of Coriolis force.

3. Centripetal force:

“This is simply the centripetal force that Moe would expect, having nothing to do with rotation (Feynman et al., 1963, section 19–4 Rotational kinetic energy).”

Unlike the previous example, an object is moving tangentially at a constant speed around the circumference of a carousel. In this example, Moe (rotating frame) observes the object moving at a velocity v_M at an instant, whereas Joe (inertial frame) sees it moving at a velocity v_J = v_M + ωr. One may add that the centripetal force is not a new or different force, but it may be a frictional force between the object and the carousel. We can provide an explanation of the direction of the Coriolis force using the cross product, but this is covered in the next chapter. Intuitively, the sidewise force is radially outward because of an inertial effect in which the object tends to be thrown out of the carousel.

Feynman explains the centripetal force on an object using the equation F_r = −mv_J²/r = −mω²r −mv_M²/r −2mv_Mω. Alternatively, from the perspective of an inertial observer, a person moving tangentially at a constant speed is essentially maintained by a frictional force (Morin, 2003). We can split the frictional force acting on the person walking on the carousel into three parts: F_friction = m(v_M + ωr)²/r = mv_M²/r + 2mv_Mωr + mω²r. The first term (mv_M²/r) is the additional frictional force on the person’s feet if he walks in a circle of radius r. The second term (2mv_Mωr) is the frictional force needed to counter the Coriolis force. The third term (mω²r) is the minimal frictional force needed that is equal to the centrifugal force due to the rotation of the carousel.

Questions for discussion:

1. How would you explain the nature of the Coriolis force?

2. How would you explain the Coriolis force for an object moving radially at a constant speed in a rotating system?

3. How would you explain the Coriolis force for an object moving tangentially (along a circumference) at a constant speed in a rotating system?

The moral of the lesson: the Coriolis force on an object is tangential when its velocity is radial, and the Coriolis force on the object is radial when its velocity is tangential.

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. Morin, D. (2003). Introductory Classical Mechanics. Cambridge: Cambridge University Press.

3. Thornton, S. T. & Marion, J. B. (2004). Classical Dynamics of Particles and Systems (5th Edition). Belmont, CA: Thomson Learning-Brooks/Cole.