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| ### Statement |
| ### Statement |
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| $7.4.35.$ [Insert problem description here] |
| $7.4.35.$ What is the period of small vibrations of four charged bodies connected by |
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| identical filaments of length l and moving as shown in the figure? Mass and |
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| charge of the body m and q. |
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| __Example Statement__: |
| __Example Statement__: |
| $1.1.1.$ Determine the coordinate $x(t)$ of a body as a function of time $t$, given that its acceleration is defined as $a(t) = bt$, where $b$ is a constant. |
| $1.1.1.$ Determine the coordinate $x(t)$ of a body as a function of time $t$, given that its acceleration is defined as $a(t) = bt$, where $b$ is a constant. |
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| ### Solution | | ### Solution |
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| [Your solution should be placed here] | | [Your solution should be placed here] |
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| __Example Solution__: | | __Example Solution__: |
| The acceleration of the body defined by | | The acceleration of the body defined by |
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| $$a(t) = bt$$ | | $$a(t) = bt$$ |
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| We know that acceleration is the time derivative of velocity: | | We know that acceleration is the time derivative of velocity: |
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| $$a(t) = \frac{d v(t)}{d t}$$ | | $$a(t) = \frac{d v(t)}{d t}$$ |
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| To find the velocity $v(t)$, we integrate $a(t)$ with respect to time: | | To find the velocity $v(t)$, we integrate $a(t)$ with respect to time: |
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| $$v(t) = \int a(t) \, dt = \int b t \, dt$$ | | $$v(t) = \int a(t) \, dt = \int b t \, dt$$ |
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| If the initial velocity is $v(0) = 0$, then the velocity becomes: | | If the initial velocity is $v(0) = 0$, then the velocity becomes: |
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| $$v(t) = \frac{b t^2}{2}$$ | | $$v(t) = \frac{b t^2}{2}$$ |
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| Likewise, integrate $v(t)$ with respect to time: | | Likewise, integrate $v(t)$ with respect to time: |
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| $$x(t)= \int v(t) \, dt = \frac{b}{2} \int t^2 \, dt$$ | | $$x(t)= \int v(t) \, dt = \frac{b}{2} \int t^2 \, dt$$ |
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| From where the coordinate from time, considering the initial conditions: | | From where the coordinate from time, considering the initial conditions: |
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| $$\boxed{x(t)=\frac{bt^3}{6}}$$ | | $$\boxed{x(t)=\frac{bt^3}{6}}$$ |
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| #### Answer | | #### Answer |
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| [Insert a concise answer or boxed result, like this:] | | [Insert a concise answer or boxed result, like this:] |
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| __Example Answer__: | | __Example Answer__: |
| $$ x(t)=\frac{bt^3}{6} $$ | | $$ x(t)=\frac{bt^3}{6} $$ |
