Section 11–6 Newton’s laws in vector notation

(Newton’s laws / Acceleration vector / Components of acceleration)

In this section, Feynman discusses Newton’s second law of dynamics, acceleration vector, and components of acceleration.

1. Newton’s laws:

“… we need not write three laws every time we write Newton’s equations or other laws of physics (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

This section could be titled as Newton’s second law of dynamics in vector notation. In a sense, Feynman seems quite sloppy when he uses the word law which means an equation. He explains the advantage of writing Newton’s laws as F = ma because this reduces the need of writing them as three laws that contain x’s, y’s, and z’s separately: F_x = m(dv_x/dt) = m(d²x/dt²) = ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z. According to Feynman, what looks like one law is really three laws for any particular set of axes because any vector equation involves the statement that each of the components is equally important. Similarly, in section 9–3 “Components of velocity, acceleration, and force,” he mentions that there are really three laws with respect to the components of the force in the x-, y-, and z-direction.

Based on a definition of acceleration in vector notations, Newton’s second law of dynamics can be written as m(d²r/dt²) = F instead of three equations using Cartesian coordinates. Feynman explains that Newton’s second law as expressed in two-dimensional polar coordinates has the advantage that we need not always write Newton’s equations as three laws. However, physics teachers should elaborate that polar coordinates are useful for problems involving radial forces and in navigation either on the sea or in the air. On the other hand, one may prefer using Cartesian coordinates to solve problems if a system has little symmetry and thus, each direction can be considered separately. In short, the advantage or disadvantage of a coordinate system is dependent on whether we can solve a problem in a convenient manner.

2. Acceleration vector:

“… to write Newton’s laws in vector form, we have to go just one step further, and define the acceleration vector (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

The acceleration vector is the time derivative of the velocity vector, a = dv/dt. The components of this vector are the second derivatives of x, y, and z with respect to time: a_x = dv_x/dt = d²x/dt², a_y = dv_y/dt = d²y/dt², and a_z = dv_z/dt = d²z/dt². Feynman also mentions that acceleration is simply Δv/Δt. However, some physicists may disagree and prefer using the term average acceleration vector and instantaneous acceleration vector. To be precise, average acceleration vector, Δv/Δt, is the difference of the velocity vectors divided by a short time interval. Furthermore, instantaneous acceleration vector, dv/dt, is measured over an infinitely small interval (or an infinitesimal). This idealized concept is defined as the limit of average acceleration vector as the time interval approaches zero.

Feynman illustrates a misconception using a figure where a particle moves at a velocity v₁ at t = t₁ and velocity v₂ at t = t₂ a little later. The average acceleration vector is the difference of the velocity vectors (v₂ and v₁) divided by a short time interval. One should not subtract the initial velocity vector, v₁, by joining the ends of v₂ and v₁ as shown in figure 11–7 “A curved trajectory.” The correct difference of the velocity vectors, Dv, is shown in figure 11–8 “Diagram for calculating the acceleration” where the tails of the vectors are joined in the same location based on the law of vector subtraction. Interestingly, the instantaneous acceleration vector (dv/dt) and the force (F) are in the same direction, but this force is not really (or ontologically) the same as the acceleration vector despite having the equality sign in the equation.

3. Components of acceleration:

“…we can think of acceleration as having two components, Δv_∥, in the direction tangent to the path and Δv_⊥ at right angles to the path (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

The acceleration vector can be conceptualized as having two perpendicular components: tangential acceleration is in the direction tangential to a particle’s path and radial acceleration is perpendicular to the path. In general, the tangential acceleration (a_∥) is a measure of how fast an object’s tangential velocity changes. Feynman states that the acceleration tangential to the path is just the change in the length of the vector and it is the change in the speed v: a_∥ = dv/dt. Specifically, the tangential acceleration can be shown as a_∥ = r(d²θ/dt²) + 2(dr/dt)(dθ/dt). The term r(d²θ/dt²) is due to the object’s changing angular velocity (or angular acceleration). The other term 2(dr/dt)(dθ/dt) is known as the Coriolis acceleration that is due to the product of the object’s radial velocity and its angular velocity.

In his Tips on Physics, Feynman adds that “it’s good to remember that formula because it’s a pain in the neck to derive it: lal = v²/r (Feynman et al., 2006, p. 46).” Mathematically, the radial acceleration of an object can be derived in only three steps: a_⊥ = Δv/Δt = vΔθ/Δt = vw (or v²/r). Firstly, we deduce the difference of the velocity vectors to be Δv = vΔθ by assuming the object’s path to be approximately circular. Secondly, the radial acceleration a_⊥ = v(Δθ/Δt) if the magnitude of the velocity (v) is constant. Thirdly, if the radius of the path is approximately r, then we deduce the angular velocity (Δθ/Δt) to be equal to v/r and conclude that a_⊥= v²/r. To be more precise, we can derive the instantaneous radial acceleration (assuming the speed is not constant) as a_⊥= d²r/dt² – v²/r.

Questions for discussion:

1. Are there three laws in the equation, F = ma?

2. How would you define an acceleration vector?

3. How would you explain the components of acceleration in a curved path of a particle?

The moral of the lesson: to determine the acceleration vector, one should understand that the difference of two velocity vectors has meaning only when the tails of the vectors coincide in the same place.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

2. 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.