<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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 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?">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<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?">
<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>
$2.4.1.$ 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="350" />
<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 $\vec{v}$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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="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 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?">
<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?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<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?">
<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?">
<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>
<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>
$2.4.1.$ 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>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="2.4.1_1.png"
<img src="2.4.1_1.png"
loading="lazy" width="350" />
loading="lazy" width="350" />
<figcaption>
<figcaption>
Relative velocity in an inertial frame of reference
Relative velocity in an inertial frame of reference
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
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 $\vec{v}$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
