Amplitude of two carts attached by a spring

[frogs]

Homework Statement

#1. [Coupled Oscillators] A cart of
mass m (car #1) and another cart of mass 2m (car #2) on a
horizontal surface are connected by an
ideal massless spring of spring constant k.
The two carts are set into oscillation about
their center-of-mass (CM) in Simple
Harmonic Motion (SHM).
(a) What is the ratio of their oscillation amplitudes A1/A2 ? Please explain your reasoning.
(b) If the CM remains at rest, what is the effective spring constant of the part of the spring (k1
and k2) between each cart and their CM in terms of k? Please give your reasoning.
(c) Let's check your answers to (b). When cart #2 moves a distance x to the right (→), how far
does cart #1 move to the left (←)? How much does the spring stretch? What is the effective
spring constant for each cart, i.e., the tension force it feels due to the spring divided by the
distance it has been displaced from its equilibrium position? Please show your calculations.
How do your findings compare with part (b)?
(d) What is the frequency of each cart's SHM oscillation? How should these frequencies
compare? Do they? Please show your calculations.
2. equations
a = -k/m • x
F = ma = -kx
1/2mv2 = 1/2kA2

although I am not sure if any of these will help

The Attempt at a Solution

[/B]for part (a) i wrote that the ratio is 2:1 because since the center of mass is 2/3 from car #1 i did 2/3:1/3.

for part (b) i did a simpaler thing that k1=2/3k and k2=1/3k

for part (c) I am stuck but i feel that if if it went x to the right the #1 car would just follow but that doesn't make sense based on the next part of c

 

[haruspex]

For part b), you need some actual logic. What you have posted appears to be a wild guess.
In part a), you found the ratio of displacements. Suppose mass 2m is displaced x relative to mass centre. What is the total extension of the spring? What is the tension in the spring? What does that tell you about the effective spring constant in relation to 2m's displacements?