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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.4">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $2.4.1.$ 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> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| </figure> | | </figure> |
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| <p> | | <p> |
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| In the initial, stationary frame of reference, the energy of the system is | | In the initial, stationary frame of reference, the energy of the system is |
| $$E_{k1} = 2\frac{mv^2}{2}$$ | | $$E_{k1} = 2\frac{mv^2}{2}$$ |
| Velocity of the system | | Velocity of the system |
| $$v_s = \frac{mv+mv}{2m} = v$$ | | $$v_s = \frac{mv+mv}{2m} = v$$ |
| In the inertial frame of reference of the system moving with velocity $\vec{v}$ | | In the inertial frame of reference of the system moving with velocity $\vec{v}$ |
| $$E_{k2} = 0 +\frac{m(2v)^2}{2}=4 \frac{mv^2}{2}$$ | | $$E_{k2} = 0 +\frac{m(2v)^2}{2}=4 \frac{mv^2}{2}$$ |
| To go to the inertial frame of reference, it is necessary to give the system of total mass $2m$ a velocity $v$ | | To go to the inertial frame of reference, it is necessary to give the system of total mass $2m$ a velocity $v$ |
| $$\Delta E_k = \frac{2mv^2}{2}$$ | | $$\Delta E_k = \frac{2mv^2}{2}$$ |
| Thus, the law of conservation of energy is fulfilled and there is no change in total kinetic energy in the reference frame | | Thus, the law of conservation of energy is fulfilled and there is no change in total kinetic energy in the reference frame |
| $$\boxed{E_{k2} = E_{k1} + \Delta E_k}$$ | | $$\boxed{E_{k2} = E_{k1} + \Delta E_k}$$ |
| </p> | | </p> |
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