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.