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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | <title>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?</title> | ||
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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <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> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.4.1_1.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | Relative velocity in an inertial frame of reference | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <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 | ||
| + | $$\boxed{E_{k2} = E_{k1} + \Delta E_k}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | In a moving frame of reference, the tension force does work. No. | ||
| + | </p> | ||
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| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | ||
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| <meta name="description" content="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?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="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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| <meta property="og:description" content="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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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <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> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.4.1_1.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| Relative velocity in an inertial frame of reference | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <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 | |||
| $$\boxed{E_{k2} = E_{k1} + \Delta E_k}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| In a moving frame of reference, the tension force does work. No. | |||
| </p> | |||
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| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
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