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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.2.$ The angular velocity of the coil is $\omega$, the radius of the inner cylinder is $r$, and the radius of the outer cylinders is $R$. What are the velocities of the coil and load axis relative to the ground? | | $1.5.2.$ The angular velocity of the coil is $\omega$, the radius of the inner cylinder is $r$, and the radius of the outer cylinders is $R$. What are the velocities of the coil and load axis relative to the ground? |
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| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.2$ | | For problem $1.5.2$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Consider the instantaneous axis of rotation passing through the point $O_1$, then the speed of the spool axis is</p> | | <p>Consider the instantaneous axis of rotation passing through the point $O_1$, then the speed of the spool axis is</p> |
| <p class="exp">$$\fbox{$v_\text{reel} = \omega R$}$$</p> | | <p class="exp">$$\fbox{$v_\text{reel} = \omega R$}$$</p> |
| <p>To find the speed of the load, consider the speed point $O$ as a superposition of the velocities of translational motion with the speed of the wheel center and the speed of rotation point $O$ relative to the wheel center.</p> | | <p>To find the speed of the load, consider the speed point $O$ as a superposition of the velocities of translational motion with the speed of the wheel center and the speed of rotation point $O$ relative to the wheel center.</p> |
| <p class="exp">$$\vec{v_O} = \vec{v}_{in} + \vec{v}_{out}$$</p> | | <p class="exp">$$\vec{v_O} = \vec{v}_{in} + \vec{v}_{out}$$</p> |
| <p class="exp">$$v_O = v_{out} – v_{in} = \omega R - \omega r = \omega\cdot (R - r)$$</p> | | <p class="exp">$$v_O = v_{out} – v_{in} = \omega R - \omega r = \omega\cdot (R - r)$$</p> |
| <p>Any point of the thread, due to its inextensibility, has the same speed, therefore, the speed of the load </p> | | <p>Any point of the thread, due to its inextensibility, has the same speed, therefore, the speed of the load </p> |
| <p class="exp">$$\fbox{$v_\text{rope} = v_O = \omega\cdot (R - r)$}$$</p> | | <p class="exp">$$\fbox{$v_\text{rope} = v_O = \omega\cdot (R - r)$}$$</p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v_\text{reel} = \omega R$$ | | $$v_\text{reel} = \omega R$$ |
| $$v_\text{cargo} = \omega (R - r)$$ | | $$v_\text{cargo} = \omega (R - r)$$ |
