Section 21–4 Initial conditions

(Define initial conditions / Determine arbitrary constants / Relate total energy)

The three important points of this section are how to define initial conditions, determine arbitrary constants, and relate the total energy of an oscillator to the amplitude of the oscillations.

1. Define initial conditions:

“The constants A and B, or a and Δ, or any other way of putting it, are determined, of course, by the way the motion started, not by any other features of the situation. These are called the initial conditions (Feynman et al., 1963, section 21–4 Initial conditions).”

Feynman says that the arbitrary constants A and B, or a and Δ, are determined by the initial conditions. He adds that the initial conditions are dependent on the way the motion was started instead of any other features of the situation. As a suggestion, one may define the initial conditions as the kinematical conditions of an object’s initial motion such as initial displacement and initial velocity that determine the subsequent motion of the object. In other words, initial conditions are dependent on the time we set to zero when we start to observe the object’s motion and how we start the motion of the object. Mathematically speaking, infinite possible solutions of a differential equation can be reduced to a unique solution when we know the initial conditions such as the initial position, initial velocity, or initial acceleration.

Feynman explains that we cannot specify the acceleration with which it started because that is determined by the spring, once we specify x₀. We may rephrase this explanation as the initial acceleration can be deduced using ma = -kx₀ and thus, it is redundant. One should clarify that it is not necessary to use the initial position and initial velocity to determine the unique solution of a second-order differential equation. Instead of using the initial position and initial velocity, we may deduce the subsequent motion of an object using the initial velocity and initial acceleration. Alternatively, we can use independent information such as the position of the object at two different times instead of the initial position and initial acceleration that are not independent.

2. Determine arbitrary constants:

“Now let us consider what determines the constants A and B, or a and Δ. Of course, these are determined by how we start the motion (Feynman et al., 1963, section 21–4 Initial conditions).”

To determine the constants A and B, we can first use the formula involving the initial position x = Acos ω₀t + Bsin ω₀t to find that x₀ = A.1 + B.0 = A. Next, we can use the formula involving the initial velocity v = −ω₀Asin ω₀t + ω₀Bcos ω₀t to find that v₀ = −ω₀A.0 + ω₀B.1 = ω₀B. Solving these two equations, we find that A = x₀ and B = v₀/ω₀. However, it is important to clarify the meaning of x₀ because some students may consider it to be the amplitude of the oscillation. Furthermore, some students may also relate v₀/ω₀ to x₀ because of the formula v = ωr. Thus, there should be a derivation of the general formula relating the velocity and displacement, v² = ω²(x_max² – x²). Using this formula, we can deduce that v₀ = ωÖ(x_max² – x₀²).

To determine the constants a and Δ, we can use the formulas involving the initial position x = a cos (ω₀t + Δ) and initial velocity v = −ω₀a sin (ω₀t + Δ). It is useful to determine a and Δ because a is the amplitude of the oscillation and it is related to the elastic potential energy (½kx²) and kinetic energy (½mv²) of the oscillator. However, Feynman did not determine the constants a and Δ of the simple harmonic motion. By specifying x = x₀ and v = v₀ at the initial instant t = 0, we can find that x₀ = a cos Δ and v₀ = −ω₀a sin Δ. Using the identity cos² Δ + sin² Δ = 1 and solving the equations, we can find that a = √(x₀² + v₀²/ω₀²) and tan Δ = −v₀/ω₀x₀. Therefore, the maximum potential energy of the oscillator can be expressed as ½ka², whereas its maximum kinetic energy can be expressed as ½ mω₀²a².

3. Relate total energy:

“The energy is dependent on the square of the amplitude; if we have twice the amplitude, we get an oscillation which has four times the energy (Feynman et al., 1963, section 21–4 Initial conditions).”

Feynman mentions that the potential energy is not constant and the potential never becomes negative. He elaborates that there is always some energy in the spring, but the amount of energy fluctuates with x. In a sense, it is sloppy to simply use the term potential energy instead of elastic potential energy. More important, one may explain that the elastic potential energy of the spring never becomes negative just like the kinetic energy because of the term x². In addition, it is not true that there is always some elastic potential energy when the spring is not extended or compressed. Based on the expression of ½kx², the elastic potential energy of a spring is usually positive when the spring is stretched or compressed, but there is no elastic potential energy if the spring is in its natural length. 

Feynman explains that the energy is dependent on the square of the amplitude of oscillation has four times the energy if the amplitude is doubled. He adds that the average elastic potential energy of the oscillator is half the maximum elastic potential energy or half the total energy. Similarly, the average kinetic energy of the oscillator is also half the total energy. Feynman did not offer an explanation of the factor of ½ using symmetrical considerations. We can determine the average kinetic energy by integrating ò½kx²dt over a period of oscillation ΔT and then dividing the integral sum by the period ΔT. As an alternative, we can explain that the graph of elastic potential energy and kinetic energy of an oscillator are both sinusoidal in shape and have the same amplitude, and thus, they must have the same average value.

Questions for discussion:

1. How would you define the initial conditions of an oscillating object?

2. How would you determine the arbitrary constants A and B, or a and Δ?

3. How would you relate the average elastic potential energy of an oscillator to its amplitude of oscillation?

The moral of the lesson: the total energy of a simple oscillator can be calculated if its initial conditions are specified.

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.