Translated 3.2.1-3.2.17
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| + | <meta name="description" content="a. A weight of mass m suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity \Omega, and point A of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is x? In what is the smallest time T that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time T/2? b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force F = -k x to the rotational motion. Determine the angular velocity of the wheel if the values of k and m are known. At what distance from the wheel axis is point A located if the largest deviation of the load from the equilibrium position is x_0?"> | ||
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| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="a. A weight of mass m suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity \Omega, and point A of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is x? In what is the smallest time T that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time T/2? b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force F = -k x to the rotational motion. Determine the angular velocity of the wheel if the values of k and m are known. At what distance from the wheel axis is point A located if the largest deviation of the load from the equilibrium position is x_0?"> | ||
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| + | <meta property="og:description" content="a. A weight of mass m suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity \Omega, and point A of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is x? In what is the smallest time T that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time T/2? b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force F = -k x to the rotational motion. Determine the angular velocity of the wheel if the values of k and m are known. At what distance from the wheel axis is point A located if the largest deviation of the load from the equilibrium position is x_0?"> | ||
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| + | <title>a. A weight of mass m suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity \Omega, and point A of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is x? In what is the smallest time T that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time T/2? b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force F = -k x to the rotational motion. Determine the angular velocity of the wheel if the values of k and m are known. At what distance from the wheel axis is point A located if the largest deviation of the load from the equilibrium position is x_0?</title> | ||
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| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
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| + | Solutions of Savchenko Problems in Physics <br> | ||
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| + | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $3.2.1.$ a. A weight of mass $m$ suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity $\Omega$, and point $A$ of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is $x$? In what is the smallest time $T$ that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time $T/2$?</p><p> | ||
| + | b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force $F = -k x$ to the rotational motion. Determine the angular velocity of the wheel if the values of $k$ and $m$ are known. At what distance from the wheel axis is point $A$ located if the largest deviation of the load from the equilibrium position is $x_0$? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | a) Equilibrium position - at the level of the wheel's center of mass | ||
| + | $$ F=-kx=\left[\omega^2=\frac{k}{m}\right]=m\omega^2x $$ | ||
| + | The values of speed and displacement of the load are repeated over time (the period of oscillation) | ||
| + | $$ t=T=\frac{2\pi}{\omega} $$ | ||
| + | The velocity vector will only change its direction, and the displacement will change sign</p><p> | ||
| + | b) Angular frequency of a spring pendulum | ||
| + | $$ \omega=\sqrt{\frac{k}{m}} $$ | ||
| + | $R=x_0$, since point $A$ is always at the same level as the weight | ||
| + | |||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Dzikan Mikita<br> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$F=m\omega^2x;\quad T=\frac{2\pi}{\omega};\quad \omega=\sqrt{\frac{k}{m}};\quad R=x_0$$ | ||
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| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="a. A weight of mass m suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity \Omega, and point A of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is x? In what is the smallest time T that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time T/2? b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force F = -k x to the rotational motion. Determine the angular velocity of the wheel if the values of k and m are known. At what distance from the wheel axis is point A located if the largest deviation of the load from the equilibrium position is x_0?"> | |||
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| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $3.2.1.$ a. A weight of mass $m$ suspended on a spring and oscillating is placed next to a wheel rotating with angular velocity $\Omega$, and point $A$ of the wheel is at the same level as the center of mass of the weight at all times. Where is the equilibrium position of the load? What force acts on the load if its displacement from its equilibrium position is $x$? In what is the smallest time $T$ that the values of the velocity and displacement of the load are repeated? How will the values of velocity and displacement change after time $T/2$?</p><p> | |||
| b. Using the results of the previous problem, compare the oscillating motion of a weight of mass m along a straight line under the action of force $F = -k x$ to the rotational motion. Determine the angular velocity of the wheel if the values of $k$ and $m$ are known. At what distance from the wheel axis is point $A$ located if the largest deviation of the load from the equilibrium position is $x_0$? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| a) Equilibrium position - at the level of the wheel's center of mass | |||
| $$ F=-kx=\left[\omega^2=\frac{k}{m}\right]=m\omega^2x $$ | |||
| The values of speed and displacement of the load are repeated over time (the period of oscillation) | |||
| $$ t=T=\frac{2\pi}{\omega} $$ | |||
| The velocity vector will only change its direction, and the displacement will change sign</p><p> | |||
| b) Angular frequency of a spring pendulum | |||
| $$ \omega=\sqrt{\frac{k}{m}} $$ | |||
| $R=x_0$, since point $A$ is always at the same level as the weight | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$F=m\omega^2x;\quad T=\frac{2\pi}{\omega};\quad \omega=\sqrt{\frac{k}{m}};\quad R=x_0$$ | |||
| </p> | |||
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