In the given figure, $a=15 ext{ m/s}^2$ represents the total acceleration of a particle moving in the clockw

Question

In the given figure, $a=15 	ext{ m/s}^2$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius $R=2.5 	ext{ m}$ at a given instant of time. The speed of the particle is:

Accepted Answer

1. **Centripetal Acceleration Component:** The particle moves in a circle, so it possesses centripetal acceleration ($a_c$) directed towards the center. The given total acceleration $a$ has a radial component ($a_c$) and a tangential component ($a_t$). Based on the standard representation of this specific problem (NEET 2016), the total acceleration vector makes an angle of $30^{\circ}$ with the radius vector.
2. **Formula:** The radial component is $a_c = a \cos 	heta$.
3. **Speed Relationship:** Centripetal acceleration is defined as $a_c = \frac{v^2}{R}$ .
4. **Calculation:**
   $$ \frac{v^2}{R} = a \cos 30^{\circ} $$
   Substituting given values ($a=15, R=2.5$):
   $$ \frac{v^2}{2.5} = 15 	imes \frac{\sqrt{3}}{2} $$
   $$ v^2 = 2.5 	imes 15 	imes 0.866 \approx 32.475 $$
   $$ v = \sqrt{32.475} \approx 5.7 	ext{ m/s} $$

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