Mastering Physics Solutions: An Electric Ceiling Fan

An Electric Ceiling Fan

Part A = 4.26 * 10^37 kg
Part B = 2.14 * 10^7
Part C = no
Part D = 6.31 * 10^10 m
Part E = yes

Solution Below:

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity of 0.280 rev/s. The angular acceleration is 0.917 rev/s^2. Its blades form a circle of diameter 0.700 m.

Part A

Compute the angular velocity of the fan after time 0.201 s has passed.
Express your answer numerically in revolutions per second.

Use the formula v = vi + at:

v = vi + at
v = 0.280 + 0.917 * 0.201
v = 0.464317 rev/s (unrounded)
v = 0.464 rev/s

0.464 rev/s

Part B

Through how many revolutions has the blade turned in the time interval 0.201 s from Part A?
Express the number of revolutions numerically.

Since we found the ending velocity in Part A, we can use the following formula. The unrounded ending speed was 0.464317 rev/s:

vf^2 = vi^2 + 2a(Δx)
0.464^2 = 0.280^2 + 2 * 0.917 * x
0.2156 = 0.0784 + 1.834x
0.1372 = 1.834x
x = 0.0748 rev

0.0746 rev

Part C

What is the tangential speed vtan(t) of a point on the tip of the blade at time = 0.201 s?
Express your answer numerically in meters per second.

Since the diameter of the fan is 0.700 m, it’s radius is 0.350 m. This gives a circumference of (c = 2πr) 2.20 m. The blade is moving at 0.464 revolutions per second, so the tangential speed is 2.20 * 0.464 = 1.02 m/s

1.02 m/s

Part D

What is the magnitude α of the resultant acceleration of a point on the tip of the blade at time t = 0.201 s?

The total acceleration includes the tangential and centripetal accelerations:

α = sqrt(a(tan)^2 + a(cent)^2)

Using the tangential velocity from Part C, we can calculate the centripetal acceleration (a = v^2/r) as 2.9726 m/s^2. The tangential acceleration is found by converting the angular acceleration to m/s^2 (similar to what we did in Part C for velocity). The tangential acceleration is therefore: a(tan) = 2πr * a(angular), or a(tan ) = 2.20 * 0.917, which equals 2.0174 m/s^2. Now combine:

α = sqrt(a(tan)^2 + a(cent)^2)
α = sqrt(2.0174^2 + 2.9726^2)
α = sqrt(4.07 + 8.8209)
α = 3.59 m/s^2

3.59 m/s^2

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