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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | <title>A rod of mass m_1 and length l is suspended on a hinge. A small piece of plasticine of mass m_2 sticks to the middle of the rod, moving until it collides with it horizontally at a speed of V. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge.</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="../#2.7">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.7.42.$ A rod of mass $m_1$ and length $l$ is suspended on a hinge. A small piece of plasticine of mass $m_2$ sticks to the middle of the rod, moving until it collides with it horizontally at a speed of $V$. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge. | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="statement.png" | ||
| + | loading="lazy" width="160" /> | ||
| + | <figcaption> | ||
| + | For problem $2.7.42$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="draw.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | Geometry of the problem | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | Angular momentum | ||
| + | $$\left\{\begin{matrix} | ||
| + | L=m_2v\frac{l}{2} \\ | ||
| + | L=I\omega | ||
| + | \end{matrix}\right.$$ | ||
| + | Parallel axis theorem | ||
| + | $$I=I_0+m_2d^2=\frac{1}{3}m_1l^2+m_2\left(\frac{l}{2}\right)^2$$ | ||
| + | $$m_2v\frac{l}{2}=\omega\left(\frac{1}{3}m_1l^2+\frac{1}{4}m_2l^2\right)$$ | ||
| + | From where | ||
| + | $$\omega = \frac{6m_2}{4m_1+3m_2}\frac{v}{l}$$ | ||
| + | From the figure | ||
| + | $$h=\frac{l}{2}(1-\cos\alpha)$$ | ||
| + | Law of conservation of energy in rotational motion | ||
| + | $$\frac{I\omega^2}{2}=(m_1+m_2)\frac{l}{2}(1-\cos\alpha)$$ | ||
| + | Expressing $\cos\alpha$ | ||
| + | $$1-\cos\alpha=\frac{I\omega^2}{(m_1+m_2)l}$$ | ||
| + | $$\boxed{\cos\alpha=1-\frac{3m_2^2v^2}{gl(4m_1+3m_2)(m_1+m_2)}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\cos\alpha=1-\frac{3m_2^2v^2}{gl(4m_1+3m_2)(m_1+m_2)}$$ | ||
| + | </p> | ||
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| <meta name="description" content="A rod of mass m_1 and length l is suspended on a hinge. A small piece of plasticine of mass m_2 sticks to the middle of the rod, moving until it collides with it horizontally at a speed of V. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A rod of mass m_1 and length l is suspended on a hinge. A small piece of plasticine of mass m_2 sticks to the middle of the rod, moving until it collides with it horizontally at a speed of V. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge."> | |||
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| <title>A rod of mass m_1 and length l is suspended on a hinge. A small piece of plasticine of mass m_2 sticks to the middle of the rod, moving until it collides with it horizontally at a speed of V. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge.</title> | |||
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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="../#2.7">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.7.42.$ A rod of mass $m_1$ and length $l$ is suspended on a hinge. A small piece of plasticine of mass $m_2$ sticks to the middle of the rod, moving until it collides with it horizontally at a speed of $V$. Find the maximum angle of deviation of the rod from the vertical. Ignore the friction in the hinge. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="statement.png" | |||
| loading="lazy" width="160" /> | |||
| <figcaption> | |||
| For problem $2.7.42$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="draw.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| Geometry of the problem | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Angular momentum | |||
| $$\left\{\begin{matrix} | |||
| L=m_2v\frac{l}{2} \\ | |||
| L=I\omega | |||
| \end{matrix}\right.$$ | |||
| Parallel axis theorem | |||
| $$I=I_0+m_2d^2=\frac{1}{3}m_1l^2+m_2\left(\frac{l}{2}\right)^2$$ | |||
| $$m_2v\frac{l}{2}=\omega\left(\frac{1}{3}m_1l^2+\frac{1}{4}m_2l^2\right)$$ | |||
| From where | |||
| $$\omega = \frac{6m_2}{4m_1+3m_2}\frac{v}{l}$$ | |||
| From the figure | |||
| $$h=\frac{l}{2}(1-\cos\alpha)$$ | |||
| Law of conservation of energy in rotational motion | |||
| $$\frac{I\omega^2}{2}=(m_1+m_2)\frac{l}{2}(1-\cos\alpha)$$ | |||
| Expressing $\cos\alpha$ | |||
| $$1-\cos\alpha=\frac{I\omega^2}{(m_1+m_2)l}$$ | |||
| $$\boxed{\cos\alpha=1-\frac{3m_2^2v^2}{gl(4m_1+3m_2)(m_1+m_2)}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\cos\alpha=1-\frac{3m_2^2v^2}{gl(4m_1+3m_2)(m_1+m_2)}$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <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> astrosander01@gmail.com <br></small> | |||
| </p> | |||
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