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| <title>The balloon deforms when it weakly hits the wall, as shown in the figure. The maximum deformation of the balloon x is much smaller than its radius R. Ignoring the change in excess pressure \Delta p of air in the balloon and the elasticity of the shell, estimate the time of impact with the wall. The mass of the ball m.</title> | | <title>The balloon deforms when it weakly hits the wall, as shown in the figure. The maximum deformation of the balloon x is much smaller than its radius R. Ignoring the change in excess pressure \Delta p of air in the balloon and the elasticity of the shell, estimate the time of impact with the wall. The mass of the ball m.</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $3.3.8^*.$ The balloon deforms when it weakly hits the wall, as shown in the figure. The maximum deformation of the balloon $x$ is much smaller than its radius $R$. Ignoring the change in excess pressure $\Delta p$ of air in the balloon and the elasticity of the shell, estimate the time of impact with the wall. The mass of the ball $m$. | | $3.3.8^*.$ The balloon deforms when it weakly hits the wall, as shown in the figure. The maximum deformation of the balloon $x$ is much smaller than its radius $R$. Ignoring the change in excess pressure $\Delta p$ of air in the balloon and the elasticity of the shell, estimate the time of impact with the wall. The mass of the ball $m$. |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| For problem $3.3.8^*$ | | For problem $3.3.8^*$ |
| </figcaption> | | </figcaption> |
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| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| Geometry of the problem | | Geometry of the problem |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| $$y=R-x$$ | | $$y=R-x$$ |
| $$r^2=R^2-(R-x)^2$$ | | $$r^2=R^2-(R-x)^2$$ |
| $$S=\pi r^2=\pi (R^2-(R-x)^2)$$ | | $$S=\pi r^2=\pi (R^2-(R-x)^2)$$ |
| Considering $x \ll R$ | | Considering $x \ll R$ |
| $$S=\pi r^2=\pi (R^2-(R-x)^2)=2\pi Rx$$ | | $$S=\pi r^2=\pi (R^2-(R-x)^2)=2\pi Rx$$ |
| Newton's second law in differential form: | | Newton's second law in differential form: |
| $$-F=m\ddot{x}$$ | | $$-F=m\ddot{x}$$ |
| $$m\ddot{x}+F=0 \Rightarrow m\ddot{x}+2\pi Rx \cdot \Delta p=0$$ | | $$m\ddot{x}+F=0 \Rightarrow m\ddot{x}+2\pi Rx \cdot \Delta p=0$$ |
| $$\ddot{x}+\frac{2\pi R\Delta p}{m}x=0$$ | | $$\ddot{x}+\frac{2\pi R\Delta p}{m}x=0$$ |
| Recalling the differential equation of harmonic motion | | Recalling the differential equation of harmonic motion |
| $$\ddot{x}+\omega ^2x=0$$ | | $$\ddot{x}+\omega ^2x=0$$ |
| Find the cyclic frequency $\omega$ of oscillation | | Find the cyclic frequency $\omega$ of oscillation |
| $$\omega=\sqrt{\frac{2\pi R\Delta p}{m}}$$ | | $$\omega=\sqrt{\frac{2\pi R\Delta p}{m}}$$ |
| Since we're interested in half of the full period, estimated time of impact with the wall could be found as | | Since we're interested in half of the full period, estimated time of impact with the wall could be found as |
| $$t=\frac{1}{2}\frac{2\pi}{\omega}=\frac{\pi}{\omega}$$ | | $$t=\frac{1}{2}\frac{2\pi}{\omega}=\frac{\pi}{\omega}$$ |
| $$\boxed{t=\pi\sqrt{\frac{m}{2\pi R\Delta p}}}$$ | | $$\boxed{t=\pi\sqrt{\frac{m}{2\pi R\Delta p}}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$t=\pi\sqrt{m/(2\pi R\Delta p)}$$ | | $$t=\pi\sqrt{m/(2\pi R\Delta p)}$$ |
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