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.

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#1.3">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <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.
    </p>

    <h3>Solution</h3>
    <p>
        <p>We will decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$</p>

In the longitudinal direction, the acceleration $a_\parallel$ is associated with the change in speed

$$a_\parallel = k$$

In the transverse direction, the acceleration $a_\perp$ is a consequence of the centrifugal force

$$a_\perp = \frac{v^2}{r} = \frac{k^2t^2}{r}$$

Full acceleration

$$a = \sqrt{a^2_\perp + a^2_\parallel}$$

Substitute the obtained components

$$\boxed{a = \sqrt{\frac{k^4t^4}{r^2} + k^2}}$$

    <h4>Answer</h4>
    <p>
        $$a = \sqrt{\frac{k^4t^4}{r^2} + k^2}$$
    </p>


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