Question

A fan rotating with an initial angular velocity of 1000 rev/minis switched off. In 2 seconds, the angular velocity decreased to200 rev/min. Assuming the angular acceleration is constant, howmany revolutions does the blade undergo during this time?

Answer 1

To calculate the number of revolutions the fan blade makes, we first convert angular velocities to rev/s, solve for angular acceleration, and then apply the equation for angular displacement during constant acceleration to find that the fan blade undergoes approximately 6.66 revolutions.

Step-by-step explanation:

To find out how many revolutions the fan blade undergoes during the deceleration period, we can apply the equations for rotational motion with constant angular acceleration. The initial angular velocity (ωi) is 1000 revolutions per minute (rev/min) and the final angular velocity (ωf) is 200 rev/min.

Firstly, we need to convert the angular velocities from rev/min to rev/s:

• ωi = (1000 rev/min) × (1 min/60 s) = 16.67 rev/s
• ωf = (200 rev/min) × (1 min/60 s) = 3.33 rev/s

Next, we use the formula ωf = ωi + αt, where t is the time and α is the angular acceleration. We can solve for α using the given values:

α = (ωf - ωi)/t = (3.33 rev/s - 16.67 rev/s) / 2 s = -6.67 rev/s²

Once α is known, we can find the total number of revolutions (Θ) using the equation Θ = ωit + 1/2αt²:

Θ = (16.67 rev/s)(2 s) + 1/2(-6.67 rev/s²)(2 s)² = 20 rev - 13.34 rev = 6.66 rev

Therefore, the fan blade undergoes approximately 6.66 revolutions during the deceleration period.