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| <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.7">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.7.41.$ A thin rod of mass $m$ and length $l$ lies on a smooth horizontal surface. A plasticine ball of mass $m$ with a velocity $v$ perpendicular to the rod hits one of its ends and sticks to it. How much heat will be released from such an impact? | | $2.7.41.$ A thin rod of mass $m$ and length $l$ lies on a smooth horizontal surface. A plasticine ball of mass $m$ with a velocity $v$ perpendicular to the rod hits one of its ends and sticks to it. How much heat will be released from such an impact? |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.7.41_1.png" | | <img src="2.7.41_1.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| Collision of a ball with a rod | | Collision of a ball with a rod |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| After collision they will move as a unit, rotating around the centre of mass, which is at a distance $l/4$ from the contact point of the balls | | After collision they will move as a unit, rotating around the centre of mass, which is at a distance $l/4$ from the contact point of the balls |
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| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.7.41_2.png" | | <img src="2.7.41_2.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| Rotations around the centre of mass | | Rotations around the centre of mass |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Conservation of angular momentum | | Conservation of angular momentum |
| $$mv\cdot\frac{L}{4}=m\frac{\omega L}{4}\cdot\frac{L}{4}+I\omega\quad(1)$$ | | $$mv\cdot\frac{L}{4}=m\frac{\omega L}{4}\cdot\frac{L}{4}+I\omega\quad(1)$$ |
| Determening the moment of inertia, using parallel axis theorem | | Determening the moment of inertia, using parallel axis theorem |
| $$I=\frac{1}{2}mL^2+m\left(\frac{L}{4}\right)^2=\frac{7}{48}mL^2\quad(2)$$ | | $$I=\frac{1}{2}mL^2+m\left(\frac{L}{4}\right)^2=\frac{7}{48}mL^2\quad(2)$$ |
| From the $(1)$ and $(2)$ | | From the $(1)$ and $(2)$ |
| $$v=\frac{5}{6}\omega L\Rightarrow \boxed{\omega=\frac{6v}{5L}}$$ | | $$v=\frac{5}{6}\omega L\Rightarrow \boxed{\omega=\frac{6v}{5L}}$$ |
| Also, in addition to rotational motion, translational motion will be observed. From the law of conservation of momentum | | Also, in addition to rotational motion, translational motion will be observed. From the law of conservation of momentum |
| $$mv=2mu\Rightarrow u=v/2$$ | | $$mv=2mu\Rightarrow u=v/2$$ |
| $u$ — centre of mass velocity</p><p> | | $u$ — centre of mass velocity</p><p> |
| THe initial energy was as a kinetic energy | | THe initial energy was as a kinetic energy |
| $$E_0=\frac{mv^2}{2}$$ | | $$E_0=\frac{mv^2}{2}$$ |
| The total energy after the collision: | | The total energy after the collision: |
| $$E_f=\frac{2mu^2}{2}+\frac{I\omega ^2}{2}+\frac{m(\omega\cdot L/u)^2}{2}$$ | | $$E_f=\frac{2mu^2}{2}+\frac{I\omega ^2}{2}+\frac{m(\omega\cdot L/u)^2}{2}$$ |
| Conservation of energy | | Conservation of energy |
| $$E_0=E_f+Q$$ | | $$E_0=E_f+Q$$ |
| Where, $Q$ — the released energy | | Where, $Q$ — the released energy |
| $$Q=E_0-E_f$$ | | $$Q=E_0-E_f$$ |
| <div class="scroll-wrapper"> | | <div class="scroll-wrapper"> |
| $$Q=\frac{mv^2}{2}-\frac{mv^2}{2\cdot2}-\frac{7mL^2}{48}\cdot \left(\frac{6v}{5L}\right)^2\cdot \frac{1}{2}+\frac{m\cdot(6v/5L\cdot L/4)^2}{2}$$ | | $$Q=\frac{mv^2}{2}-\frac{mv^2}{2\cdot2}-\frac{7mL^2}{48}\cdot \left(\frac{6v}{5L}\right)^2\cdot \frac{1}{2}+\frac{m\cdot(6v/5L\cdot L/4)^2}{2}$$ |
| </div> | | </div> |
| After the mathematical transformations | | After the mathematical transformations |
| $$\boxed{Q=\frac{mv^2}{10}}$$ | | $$\boxed{Q=\frac{mv^2}{10}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$Q=1/10\cdot mv^2$$ | | $$Q=1/10\cdot mv^2$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
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