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| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
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| $1.4.13.$ A gear of radius $R$ is placed between two parallel gear racks. The slats move at a speed of $v_1$ and $v_2$ towards each other. What is the gear speed | | $1.4.13.$ A gear of radius $R$ is placed between two parallel gear racks. The slats move at a speed of $v_1$ and $v_2$ towards each other. What is the gear speed |
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| For problem $1.4.13$ | | For problem $1.4.13$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Let's move to the center of the disk. In this frame of reference, the speed of the upper and lower points of the disk is the same in magnitude and opposite in direction. Therefore, </p> | | <p>Let's move to the center of the disk. In this frame of reference, the speed of the upper and lower points of the disk is the same in magnitude and opposite in direction. Therefore, </p> |
| <p class="exp">$$(v_{1} - v) = - (v_{2} - v)$$</p> | | <p class="exp">$$(v_{1} - v) = - (v_{2} - v)$$</p> |
| | | |
| <p>From where</p> | | <p>From where</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ v = \frac{v_{1} + v_{2}}{2} $$ | | $$ v = \frac{v_{1} + v_{2}}{2} $$ |
| </p> | | </p> |
| <p>We will find the rotation frequency as </p> | | <p>We will find the rotation frequency as </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \nu = \frac{v}{R} $$ | | $$ \nu = \frac{v}{R} $$ |
| </p> | | </p> |
| <p>Substitute $v$</p> | | <p>Substitute $v$</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \fbox{$\nu = \frac{v_1 + v_2}{2R}$} $$ | | $$ \fbox{$\nu = \frac{v_1 + v_2}{2R}$} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$n = (v_1 + v_2)/(2R)$$ | | $$n = (v_1 + v_2)/(2R)$$ |
| </p> | | </p> |
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