Chapter 40, Problem 30P

Two particles with masses m₁ and m₂ are joined by a light spring of force constant k. They vibrate along a straight line with their center of mass fixed. (a) Show that the total energy

$\frac{1}{2}{m}_1{u}_1^2+\frac{1}{2}{m}_2{u}_2^2+\frac{1}{2}k{x}^2$

can be written as $\frac{1}{2}\mu u^2+\frac{1}{2}k{x}^2$, where u = |u₁| + |u₂| is the relative speed of the particles and μ = m₁m₂/(m₁ + m₂) is the reduced mass of the system. This result demonstrates that the pair of freely vibrating particles can be precisely modeled as a single particle vibrating on one end of a spring that has its other end fixed. (b) Differentiate the equation

$\frac{1}{2}\mu u^2+\frac{1}{2}k{x}^2=\text{constant}$

with respect to x. Proceed to show that the system executes simple harmonic motion. (c) Find its frequency.