special relativity - thought experiment concerning $E = mcdot c^2$

Setup:

Suppose one has two identical wheels $W_1$ and $W_2$. Wheel $W_1$ is rotating about its axis with angular velocity $\vec{\omega}$ while the other wheel is not rotating. Imagine then two identical carts $C_1$ and $C_2$ with the rotating wheel $W_1$ inside $C_1$ and the non-rotating wheel $W_2$ inside the cart $C_2$. Initially the velocity of both carts is $\vec{v}_1 = \vec{v_2} = \begin{bmatrix}0 &0 &0 \end{bmatrix}^T$.

Question:

Suppose that to both carts $C_1$ and $C_2$ is applied the same constant force $\vec{F}$ for $1$ second. After $1$ second is the velocity of cart $C_1$ slightly smaller then the velocity of cart $C_2$?

Answer (...)

This is a question that I formed to test some understanding about relativity. I think the wheel $W_1$ has a greater energy then the wheel $W_2$ hence it has a slightly greater inertial mass, therefore the final velocity of cart $C_1$ should be smaller then the velocity of cart $C_2$! Is this correct?

I have never worked seriously with relativity, but if I take the notorious formula $E = m\cdot c^2$ and consider $E_1 = \frac{1}{2} \cdot I\cdot \omega^2 > 0 = E_2$ follows that the first wheel has the total energy $E_{1t} = m\cdot c^2 + E_1$ before the cart was moving while and the second wheel has the total energy $E_{2t} = m\cdot c^2$. When the the force is applied to the first cart, it tries to move a mass $m_1 = m + \frac{E_1}{c^2} > m$ hence cart $C_1$ should have a smaller acceleration. PS: The second cart will gradually increase its translational energy hence its mass should also be increased, but assume the rotational energy of the former is much greater ... Is this reasoning correct?

Answer

Yes, this is correct. Energy increases inertia. Of course, in typical situations $E_1/c^2$ is much much smaller than the masses, which is why this effect was discovered theoretically and not experimentally.