First slide

The two types of acceleration

Question

A particle moves in a circular path such that its speed v varies with distance as v = α√s where $\alpha$ is a positive constant. Find the acceleration of particle after traversing a distance S?

Difficult

A

$\large {\alpha ^2}\sqrt {\frac{1}{4} - \frac{{{S^2}}}{{{R^2}}}}$
B

${\alpha ^2}\sqrt {\frac{1}{4} + \frac{{{S^2}}}{{{R^2}}}}$
C

$\alpha \sqrt {\frac{1}{2} + \frac{{{S^2}}}{{{R^2}}}}$
D

${\alpha ^2}\sqrt {\frac{1}{2} + \frac{{{S^2}}}{{{R^2}}}}$

Solution

$\large \nu = \alpha \sqrt s ,{a_t} = \frac{{d\nu }}{{dt}} = \frac{{d\nu }}{{ds}}\frac{{ds}}{{dt}} \Rightarrow {a_t} = v\frac{{d\nu }}{{ds}}\\\\= (\alpha \sqrt s )[ {(\alpha )\frac{1}{2}\frac{1}{{\sqrt s }}} = \frac{{{\alpha ^2}}}{2}\\\\{{a_c} = \frac{{{v^2}}}{R} = \frac{{{\alpha ^2}s}}{R} = {\alpha ^2}\sqrt {\frac{1}{4} + \frac{{{s^2}}}{{{R^2}}}}}$
$a = \sqrt{a_{r}^{2} + a_{t}^{2}}$

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