How Many Revolutions Does the Blade Undergo?

What is angular acceleration?

Angular acceleration is defined as the pace of change of angular velocity with reference to time.

Angular acceleration calculation:

Given initial angular velocity = 1000 rev/min = 16.67 rev/sec.

Final angular velocity = 200 rev/min = 3.33 rev/sec.

Time taken = 2 seconds.

Average angular acceleration formula: [tex]\alpha_{\text{avg}} = \frac{w_f - w_i}{t_f - t_i}[/tex]

Calculating angular acceleration:

[tex]\alpha_{\text{avg}} = \frac{3.33 - 16.67}{2} = -6.67 \text{ rev/sec}^2[/tex]

Angular acceleration is a measure of the rate at which the angular velocity of an object changes over time. In this scenario, the angular acceleration of the fan was calculated to be -6.67 rev/sec^2.

When the fan was switched off, the angular velocity decreased from 1000 rev/min to 200 rev/min over 2 seconds. This change in velocity, combined with the constant angular acceleration, allows us to determine the number of revolutions the blade undergoes.

Using the formula [tex]\theta = \omega_i t + \frac{1}{2} \alpha t^2[/tex], where [tex]\omega_i[/tex] is the initial angular velocity, [tex]\alpha[/tex] is the angular acceleration, and [tex]t[/tex] is the time, we can calculate the number of revolutions the blade experiences.

Substituting the values:

[tex]\theta = 16.67 \times 2 + \frac{1}{2}(-6.67) \times 2^2 = 20 \text{ rev}[/tex]

Hence, the blade undergoes 20 revolutions during this time period.