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| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $3.2.9.$ By how much will a pendulum clock raise to the height of Everest ($8.9$ km) lag behind in a day? Ostankino Tower ($0.5$ km)? | | $3.2.9.$ By how much will a pendulum clock raise to the height of Everest ($8.9$ km) lag behind in a day? Ostankino Tower ($0.5$ km)? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| The period of oscillation of a mathematical pendulum | | The period of oscillation of a mathematical pendulum |
| $$ T_0=2\pi\sqrt{\frac{l}{g}}; \quad T_1=2\pi\sqrt{\frac{l}{g^*}} $$ | | $$ T_0=2\pi\sqrt{\frac{l}{g}}; \quad T_1=2\pi\sqrt{\frac{l}{g^*}} $$ |
| Acceleration of gravity depending on the distance to the center of the Earth of mass $M$ | | Acceleration of gravity depending on the distance to the center of the Earth of mass $M$ |
| $$ g=\frac{GM}{R^2} $$ | | $$ g=\frac{GM}{R^2} $$ |
| | | |
| $$ g^*=\frac{GM}{(R+H)^2} $$ | | $$ g^*=\frac{GM}{(R+H)^2} $$ |
| The ratio of accelerations of gravity for different distances to the center | | The ratio of accelerations of gravity for different distances to the center |
| $$ \frac{g}{g^*}=\frac{(R+H)^2}{R^2}\Rightarrow g^*=\frac{gR^2}{(R+H)^2} $$ | | $$ \frac{g}{g^*}=\frac{(R+H)^2}{R^2}\Rightarrow g^*=\frac{gR^2}{(R+H)^2} $$ |
| Using the approximation for a small value of $x =\frac{h}{R} \ll 1$; $(1+x)^\alpha\approx 1+\alpha x$: | | Using the approximation for a small value of $x =\frac{h}{R} \ll 1$; $(1+x)^\alpha\approx 1+\alpha x$: |
| $$ T_1=2\pi\frac{R+H}{R}\sqrt{\frac{l}{g}} $$ | | $$ T_1=2\pi\frac{R+H}{R}\sqrt{\frac{l}{g}} $$ |
| From where we find the required lag as | | From where we find the required lag as |
| $$ \Delta T_1=T_1-T_0=T_0(\frac{R+H}{R}-1)=2\text{ min} $$ | | $$ \Delta T_1=T_1-T_0=T_0(\frac{R+H}{R}-1)=2\text{ min} $$ |
| | | |
| $$ \Delta T_2=T_2-T_0=T_0(\frac{R+h}{R}-1)=6.75\text{ s} $$ | | $$ \Delta T_2=T_2-T_0=T_0(\frac{R+h}{R}-1)=6.75\text{ s} $$ |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Dzikan Mikita<br> | | Dzikan Mikita<br> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\Delta T_1=2\text{ min};\quad\Delta T_2=6.75\text{ s}$$ | | $$\Delta T_1=2\text{ min};\quad\Delta T_2=6.75\text{ s}$$ |
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