A particle completes 3 revolutions per second on a circular path of radius 8 cm. Find the values of angular velocity and centripetal acceleration of the particle -

Options

A. `6pi ("rad")/(s) , 288 pi^(2) (cm)/(s^(2))`
B. `pi ("rad")/(s) , 275 pi^(2) (cm)/(s^(2))`
C. `6pi ("rad")/(s) , 288 (cm)/(s^(2))`
D. none

Answer

To solve the problem, we will calculate the angular velocity and centripetal acceleration of a particle moving in a circular path.

**Step 1: Calculate the angular velocity (ω)**

Given:
- The particle completes 3 revolutions per second (f = 3 rev/s).

The formula for angular velocity (ω) in radians per second is:
\[ \omega = 2\pi f \]

Substituting the value of f:
\[ \omega = 2\pi \times 3 = 6\pi \text{ radians/second} \]

**Step 2: Calculate the centripetal acceleration (a_c)**

Given:
- The radius of the circular path (r) = 8 cm = 0.08 m (convert cm to m for standard SI units).

The formula for centripetal acceleration (a_c) is:
\[ a_c = \frac{v^2}{r} \]
where v is the linear velocity. We can find v using the relationship between linear velocity and angular velocity:
\[ v = \omega r \]

Substituting the values:
\[ v = (6\pi) \times (0.08) = 0.48\pi \text{ m/s} \]

Now, substituting v back into the centripetal acceleration formula:
\[ a_c = \frac{(0.48\pi)^2}{0.08} \]

Calculating \( (0.48\pi)^2 \):
\[ (0.48\pi)^2 = 0.2304\pi^2 \]

Now substituting this value into the centripetal acceleration formula:
\[ a_c = \frac{0.2304\pi^2}{0.08} = 2.88\pi^2 \text{ m/s}^2 \]

To convert this back to cm/s²:
\[ a_c = 2.88\pi^2 \times 100 \text{ cm/s}^2 = 288\pi^2 \text{ cm/s}^2 \]

**Final Results:**
- Angular velocity (ω) = \( 6\pi \) radians/second
- Centripetal acceleration (a_c) = \( 288\pi^2 \) cm/s²

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