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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.3.25.$ A small body moves along a circle of radius $r$ at a speed that increases linearly in time according to the law $v = kt$. Find the dependence of the total acceleration of the body on time. | | $1.3.25.$ A small body moves along a circle of radius $r$ at a speed that increases linearly in time according to the law $v = kt$. Find the dependence of the total acceleration of the body on time. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>We will decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$</p> | | <p>We will decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$</p> |
| <p>In the longitudinal direction, the acceleration $a_\parallel $ is associated with the change in speed </p> | | <p>In the longitudinal direction, the acceleration $a_\parallel $ is associated with the change in speed </p> |
| <p class="exp"> | | <p class="exp"> |
| $$a_\parallel = k$$ | | $$a_\parallel = k$$ |
| </p> | | </p> |
| <p>In the transverse direction, the acceleration $a_\perp $ is a consequence of the centrifugal force</p> | | <p>In the transverse direction, the acceleration $a_\perp $ is a consequence of the centrifugal force</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ a_\perp = \frac{v^2}{r} = \frac{k^2t^2}{r} $$ | | $$ a_\perp = \frac{v^2}{r} = \frac{k^2t^2}{r} $$ |
| </p> | | </p> |
| <p>Full acceleration</p> | | <p>Full acceleration</p> |
| <p class="exp"> | | <p class="exp"> |
| $$a = \sqrt{a^2_\perp + a^2_\parallel}$$ | | $$a = \sqrt{a^2_\perp + a^2_\parallel}$$ |
| </p> | | </p> |
| <p>Substitute the obtained components</p> | | <p>Substitute the obtained components</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \boxed{a = \sqrt{\frac{k^4t^4}{r^2} + k^2}} $$ | | $$ \boxed{a = \sqrt{\frac{k^4t^4}{r^2} + k^2}} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$a = \sqrt{\frac{k^4t^4}{r^2} + k^2}$$ | | $$a = \sqrt{\frac{k^4t^4}{r^2} + k^2}$$ |
| </p> | | </p> |
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