Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.300 rev/s . The magnitude of the angular acceleration is 0.920 rev/s2 . Both the the angular velocity and angular acceleration are directed counterclockwise. The electric ceiling fan blades form a circle of diameter 0.740 m .

Part A:

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

Express your answer numerically in revolutions per second (rev/s)

Part B:

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

Express the number of revolutions numerically.

Part C:

What is the tangential speed v_t of a point on the tip of the blade at time ttt = 0.191 ss ?

Express your answer numerically in meters per second.

Part D:

Calculate the magnitude a_t of the tangential acceleration of a point on the tip of the blade at time ttt = 0.191 ss .

Express the acceleration numerically in meters per second squared.

Part E:

Calculate the magnitude a_r of the radial (or centripetal) acceleration of the point at the end of the fan blade.

Answer 1

Given the initial angular velocity = 0.300 rev/s,  angular acceleration = 0.920 rev/s^2 and the diameter of the circle d = 0.740m. So the radius of the circle is r = 0.370m.

(A) At t = 0.191s, the angular velocity of the fan is given by,

So the angular velocity of the fan after t = 0.191s is 0.47rev/s.

(B) The nunber of revolutions is given by,

So the blade turns 0.07 revolutions in the mentioned time interval.

(C) The angular velocity at 0.191s,

The tangential speed of a point on the tip of the blade at 0.191s is,

So the tangential speed of the point on the tip of the blade at 0.191s is 1.09m/s.

(D) Given the angular acceleration,

The tangential acceleration of a point on the tip of the blade at 0.191s is

So the tangential acceleration of the point on the tip of the blade at 0.191s is 2.14m/s^2.

(E) The radial acceleration is given by,

So the centripetal acceleration of the point at the end of the fan blade is 3.21m/s^2.