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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> |
| <p> | | <p> |
| $2.1.36.$ The velocity of a body of mass $m$ in a viscous liquid decreases with the distance $l$ traveled according to the law $v = v_0 - \beta l$, where $v_0$ is the initial velocity, and $\beta$ is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body? | | $2.1.36.$ The velocity of a body of mass $m$ in a viscous liquid decreases with the distance $l$ traveled according to the law $v = v_0 - \beta l$, where $v_0$ is the initial velocity, and $\beta$ is a constant coefficient. How does the viscous friction force acting on a body from the fluid side depend on the velocity of the body? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| The equation of Newton's second law for the direction of motion: | | The equation of Newton's second law for the direction of motion: |
| $$ ma = F_с $$ | | $$ ma = F_с $$ |
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| $$ \frac{F_с}{m}=\frac{dv_x}{dt} $$ | | $$ \frac{F_с}{m}=\frac{dv_x}{dt} $$ |
| Derivative of velocity with respect to time | | Derivative of velocity with respect to time |
| $$ \frac{dv_x}{dt}=\frac{d}{dt}(v_0-βx) $$ | | $$ \frac{dv_x}{dt}=\frac{d}{dt}(v_0-βx) $$ |
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| $$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-β\frac{dx}{dt}=-βt $$ | | $$ \frac{dv_x}{dt}=\frac{dv_0}{dt}-β\frac{dx}{dt}=-βt $$ |
| the minus sign shows that the acceleration vector is directed in the direction opposite to the velocity vector. | | the minus sign shows that the acceleration vector is directed in the direction opposite to the velocity vector. |
| Combining the equations, we obtain the value of the resistance force as a function of velocity | | Combining the equations, we obtain the value of the resistance force as a function of velocity |
| $$ \boxed{F=βmv} $$ | | $$ \boxed{F=βmv} $$ |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
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| $$F = βmv$$ | | $$F = βmv$$ |
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