Final answer:
To find the required rotational speed in rpm, use the formula for centripetal acceleration and solve for angular velocity. Given the values of distance and acceleration, calculate the angular velocity and convert it to rpm. The answer is approximately 720 rpm. so, option d is the correct answer.
Step-by-step explanation:
To determine the required rotational speed (in rpm) at which a particle 8.6 cm from the axis of rotation experiences an acceleration of 500,000 g's, we can use the formula for centripetal acceleration:
ac = rω2
Where ac is the centripetal acceleration, r is the distance from the axis of rotation, and ω is the angular velocity.
Given that ac = 500,000g and r = 8.6 cm, we need to solve for ω. Rearranging the formula, we have:
ω = √(ac/r)
Substituting the given values, we get:
ω = √(500,000g/8.6 cm)
Converting to the appropriate units, 1 g ≈ 9.8 m/s² and 1 cm = 0.01 m, the equation becomes:
ω = √(500,000 * 9.8 m/s² / 0.086 m)
Calculating the value gives:
ω ≈ 610 rad/s
To convert this to rpm, we can use the conversion factor: 1 rev = 2π rad. Therefore:
ω (in rpm) ≈ (610 rad/s * 60 s) / (2π rad)
Calculating this gives:
ω (in rpm) ≈ 610 * 60 / (2π)
Finally, rounding to two significant figures, the answer is approximately 720 rpm.