An electric ceiling fan is rotating about a fixed axis with an initial angular v

Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.270 rev/s . The magnitude of the angular acceleration is 0.892 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.700 m .

Part A

Compute the fan's angular velocity magnitude after time 0.191 s has passed.

Express your answer numerically in revolutions per second.

Part B

Through how many revolutions has the blade turned in the time interval 0.191 s from Part A?

Express the number of revolutions numerically.

Part C

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.191 s ?

Express your answer numerically in meters per second.

Part D
What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.191 s ?

Express the acceleration numerically in meters per second squared.

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.270 rev/s . The magnitude of the angular acceleration is 0.892 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.700 m .

Part A

Compute the fan's angular velocity magnitude after time 0.191 s has passed.

Express your answer numerically in revolutions per second.

Part B

Through how many revolutions has the blade turned in the time interval 0.191 s from Part A?

Express the number of revolutions numerically.

Part C

What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.191 s ?

Express your answer numerically in meters per second.

Part D
What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.191 s ?

Express the acceleration numerically in meters per second squared.

Explanation / Answer

given,

angular velocity = 0.27 rev/sec or 1.69646 rad/sec

angular acceleration = 0.892 rev/sec^2 or 5.6046 rad/sec^2

diameter = 0.7 m

by first equation of motion

v = u + at

v = 1.69646 + 5.6046 * 0.191

v = 2.7669 rad/sec

2 pi rad = 1 rev so,

fan's angular velocity = 2.7669 rad/sec or 0.4403 rev

by third equation of motion

v^2 = u^2 + 2as

2.7669^2 = 0 + 2 * 5.6046 * s

s = 85.8146 rads

2 pi rad = 1 rev so,

distance s = 13.65781778 revolution

tangential velocity = angular velocity * radius

tangential velocity = 2.7669 * (0.7 / 2)

tangential velocity = 0.9684 m/s

tangential acceleration = angular acceleration * radius

tangential acceleration = 5.6046 * (0.7 / 2)

tangential acceleration = 1.96161 m/s^2

radial acceleration = v^2 / r

radial acceleration = 0.9684^2 / (0.7/2)

radial acceleration = 2.6794 m/s^2

resultant acceleration = sqrt(radial acceleration^2 + tangential acceleration^2)

resultant acceleration = sqrt(2.6794^2 + 1.96161^2)

resultant acceleration = 3.32 m/s^2

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