Question

A centrifuge whose maximum rotation rate is ? = 10, 000 revolutions per minute (rpm) can be brought to rest in a time ?t = 97.6 s. Assume constant angular acceleration. (a) What is the angular speed, in SI units, just before it begins decelerating? (b) What is the angular acceleration in SI units? (c) How far (m) does a point R = 8.13 cm from the center travel during the deceleration? (d) What is the radial component of acceleration (m/s2) of the point just as the centrifuge begins its deceleration? (e) What is the tangential component of acceleration (m/s2) of the point just as the centrifuge begins its deceleration?

Answer 1

The angular speed just before the centrifuge begins decelerating is approximately 1047.2 rad/s. The angular acceleration is approximately -10.72 rad/s². A point 8.13 cm from the center of the centrifuge travels approximately 473.53 m during the deceleration. The radial component of acceleration is approximately -0.87 m/s², and the tangential component of acceleration is also approximately -0.87 m/s².

Step-by-step explanation:

(a) To convert revolutions per minute (rpm) to radians per second (rad/s), we use the formula:

angular speed (rad/s) = (angular speed (rpm) * 2π) / 60

Given that the maximum rotation rate is 10,000 rpm, the angular speed just before it begins decelerating is:

angular speed = (10,000 * 2π) / 60 ≈ 1047.2 rad/s

(b) The angular acceleration can be calculated using the formula:

angular acceleration = (final angular speed - initial angular speed) / time

Since the centrifuge comes to rest, the final angular speed is 0, the initial angular speed is 1047.2 rad/s, and the time is 97.6 s, the angular acceleration is:

angular acceleration = (0 - 1047.2) / 97.6 ≈ -10.72 rad/s²

(c) The distance traveled by a point R = 8.13 cm from the center during the deceleration can be calculated using the formula:

distance = (initial angular speed * time) + (0.5 * angular acceleration * time^2)

Substituting the known values, we have:

distance = (1047.2 * 97.6) + (0.5 * -10.72 * (97.6)^2) ≈ 47352.65 cm ≈ 473.53 m

(d) The radial component of acceleration can be determined using the formula:

radial acceleration = (angular acceleration) * (distance from the center)

Substituting the known values, we have:

radial acceleration = (-10.72) * 0.0813 ≈ -0.87 m/s²

(e) The tangential component of acceleration is given by the formula:

tangential acceleration = (angular acceleration) * (distance from the center)

Substituting the known values, we have:

tangential acceleration = (-10.72) * 0.0813 ≈ -0.87 m/s²

Answer 2

The angular speed just before the centrifuge begins decelerating is 1047.2 rad/s. The angular acceleration is -10.72 rad/s². A point 8.13 cm from the center travels 8.309 m during the deceleration. The radial component of acceleration is -0.869 m/s² and the tangential component of acceleration is -0.869 m/s².

Step-by-step explanation:

To answer the given questions:

(a) To find the angular speed just before decelerating, we need to convert the maximum rotation rate from rpm to rad/s. The conversion factor is 1 revolution = 2π radians. Therefore, the angular speed just before decelerating is (10000 rpm)(2π rad/1 min)(1 min/60 s) = 1047.2 rad/s.

(b) The angular acceleration can be calculated using the formula angular acceleration = (final angular speed - initial angular speed) / time. From the given information, the initial angular speed is 1047.2 rad/s, the final angular speed is 0 rad/s, and the time is 97.6 s. Plugging these values into the formula gives an angular acceleration of -10.72 rad/s². The negative sign indicates deceleration.

(c) The distance traveled by a point at a radius of 8.13 cm can be calculated using the formula distance = (initial angular speed)(time) + (1/2)(angular acceleration)(time²). Plugging in the values, we get a distance of 830.9 cm or 8.309 m.

(d) The radial component of acceleration can be calculated using the formula radial acceleration = (radius)(angular acceleration). In this case, the radius is 8.13 cm or 0.0813 m, and the angular acceleration is -10.72 rad/s². Plugging in these values gives a radial acceleration of -0.869 m/s².

(e) The tangential component of acceleration can be calculated using the formula tangential acceleration = (radius)(angular acceleration). Using the same values as in part (d), we get a tangential acceleration of -0.869 m/s².