The angular speed of the rotor in a centrifuge decreases from 226 to 186.0 rev/s with an angular deceleration of 6.4 rev/s2. During this period, how many revolutions does the
rotor turn?
N=(n₂²-n₁²)/2a=(186²-226²)/(-2•6.4)=
=1287.5 rev
To find the number of revolutions the rotor turns during the period, we need to calculate the average angular speed of the rotor and then use it to determine the time it takes to decelerate from 226 to 186.0 rev/s.
First, we can calculate the average angular speed by finding the average of the initial and final angular speeds:
Average angular speed = (Initial angular speed + Final angular speed) / 2
= (226 rev/s + 186.0 rev/s) / 2
= 206 rev/s
Next, we can use the formula of angular acceleration to find the time it takes to decelerate:
Angular acceleration = (Final angular speed - Initial angular speed) / time
Rearranging the formula:
time = (Final angular speed - Initial angular speed) / Angular acceleration
= (186.0 rev/s - 226 rev/s) / -6.4 rev/s^2
= -40.0 rev/s / -6.4 rev/s^2
= 6.25 s
Finally, we can calculate the number of revolutions using the equation:
Number of revolutions = Average angular speed * time
= 206 rev/s * 6.25 s
= 1287.5 revolutions
Therefore, the rotor turns approximately 1287.5 revolutions during the period.
To find the total number of revolutions the rotor turns during the given time period, we can use the kinematic equation for angular motion:
ω^2 = ω_0^2 + 2αθ
Where:
ω is the final angular velocity (in rad/s),
ω_0 is the initial angular velocity (in rad/s),
α is the angular acceleration (in rad/s^2),
and θ is the angular displacement (in radians).
First, let's convert the given quantities into radians:
ω_0 = 226 rev/s = 226 * 2π rad/s ≈ 1420.53 rad/s
ω = 186.0 rev/s = 186.0 * 2π rad/s ≈ 1168.09 rad/s
α = -6.4 rev/s^2 = -6.4 * 2π rad/s^2 ≈ -40.21 rad/s^2
Now, we can solve for θ:
θ = (ω^2 - ω_0^2) / (2α)
θ = (1168.09^2 - 1420.53^2) / (2 * -40.21)
θ ≈ -66,941.46 rad
Since we're interested in the absolute value of the angular displacement, we take the magnitude:
θ ≈ 66,941.46 rad
Now, we can convert the angular displacement into the number of revolutions:
Number of revolutions = θ / (2π)
Number of revolutions ≈ 66,941.46 / (2π)
Number of revolutions ≈ 10,659.77 revolutions
Therefore, during this period, the rotor turns approximately 10,659.77 revolutions.