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| <title>Determine the steady-state velocity of a body on an inclined plane that changes one direction of its velocity u to the opposite direction with high frequency. The direction of movement of the plane is shown in the figure. Coefficient of friction \mu, angle of inclination of the plane \alpha, tg \alpha < \mu.</title> | | <title>Determine the steady-state velocity of a body on an inclined plane that changes one direction of its velocity u to the opposite direction with high frequency. The direction of movement of the plane is shown in the figure. Coefficient of friction \mu, angle of inclination of the plane \alpha, tg \alpha < \mu.</title> |
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| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
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| $2.1.44^*.$ Determine the steady-state velocity of a body on an inclined plane that changes one direction of its velocity $u$ to the opposite direction with high frequency. The direction of movement of the plane is shown in the figure. Coefficient of friction $\mu$, angle of inclination of the plane $\alpha$, $tg \alpha < \mu$. | | $2.1.44^*.$ Determine the steady-state velocity of a body on an inclined plane that changes one direction of its velocity $u$ to the opposite direction with high frequency. The direction of movement of the plane is shown in the figure. Coefficient of friction $\mu$, angle of inclination of the plane $\alpha$, $tg \alpha < \mu$. |
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| For problem $2.1.44^*$ | | For problem $2.1.44^*$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the body | | Forces acting on the body |
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| Since the speed changes quickly, the body does not have time to move in the horizontal direction and all the time moves in the direction of $\vec{v}$ | | Since the speed changes quickly, the body does not have time to move in the horizontal direction and all the time moves in the direction of $\vec{v}$ |
| </p> | | </p> |
| <p> | | <p> |
| Since $v=\text{const}$, there is no acceleration in the direction along the speed $\vec{v}$ | | Since $v=\text{const}$, there is no acceleration in the direction along the speed $\vec{v}$ |
| $$ mg\sin\alpha = \mu mg \cos\alpha\cos\beta $$ | | $$ mg\sin\alpha = \mu mg \cos\alpha\cos\beta $$ |
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| $$ \cos\beta = \frac{\tan\alpha}{\mu} $$ | | $$ \cos\beta = \frac{\tan\alpha}{\mu} $$ |
| From geometric considerations, the moduli of the vectors $\vec{v}$ and $\vec{u}$ are related by the relation | | From geometric considerations, the moduli of the vectors $\vec{v}$ and $\vec{u}$ are related by the relation |
| $$ v=\frac{u}{\tan\beta} $$ | | $$ v=\frac{u}{\tan\beta} $$ |
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| $$ v=u \frac{\frac{\tan\alpha}{\mu}}{\sqrt{1-\frac{\tan^2\alpha}{\mu^2}}} $$ | | $$ v=u \frac{\frac{\tan\alpha}{\mu}}{\sqrt{1-\frac{\tan^2\alpha}{\mu^2}}} $$ |
| | | |
| $$ \boxed{v = u \frac{\tan α}{\sqrt{\mu ^{2} − \tan^{2} α}}} $$ | | $$ \boxed{v = u \frac{\tan α}{\sqrt{\mu ^{2} − \tan^{2} α}}} $$ |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
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| $$v = u \frac{\, \text{tg} α}{\sqrt{\mu ^{2} − \text{tg}^{2} α}}$$ | | $$v = u \frac{\, \text{tg} α}{\sqrt{\mu ^{2} − \text{tg}^{2} α}}$$ |
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