Balls of mass m, each connected by a thread, move in a circle with a constant velocity v. The kinetic energy of each ball, equal to \frac{mv^2}{2}, does not change. If we switch to a reference frame in which the center of the thread moves in a straight line in the plane of rotation at a speed , the energy of each of the balls changes from zero to 4 \frac{mv^2}{2}. What causes this energy change? Does the total kinetic energy change in the specified reference frame?

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#2.4">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $2.4.1.$ Balls of mass $m$, each connected by a thread, move in a circle with a constant velocity $v$. The kinetic energy of each ball, equal to $\frac{mv^2}{2}$, does not change. If we switch to a reference frame in which the center of the thread moves in a straight line in the plane of rotation at a speed $,$ the energy of each of the balls changes from zero to $4 \frac{mv^2}{2}$. What causes this energy change? Does the total kinetic energy change in the specified reference frame?
    </p>

    <h3>Solution</h3>
    <p>

In the initial, stationary frame of reference, the energy of the system is
$$E_{k1} = 2\frac{mv^2}{2}$$
Velocity of the system
$$v_s = \frac{mv+mv}{2m} = v$$
In the inertial frame of reference of the system moving with velocity $\vec{v}$
$$E_{k2} = 0 +\frac{m(2v)^2}{2}=4 \frac{mv^2}{2}$$
To go to the inertial frame of reference, it is necessary to give the system of total mass $2m$ a velocity $v$
$$\Delta E_k = \frac{2mv^2}{2}$$
Thus, the law of conservation of energy is fulfilled and there is no change in total kinetic energy in the reference frame
$$\boxed{E_{k2} = E_{k1} + \Delta E_k}$$

    <h4>Answer</h4>
    <p>
        In a moving frame of reference, the tension force does work. No.
    </p>
    <p style="text-align: right; font-style: italic; font-size: 14;">
      Almaskhan Arsen<br>
    </p>


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