The spin-drier of a washing machine slows down uniformly from 700 rpm (revolutions per minute) to 140 rpm while making 50 revolutions. Find the angular acceleration through these 50 revolutions. Express your answer in rad/s^2.
(140-700) = -560 rev/min
-560rev/min * 1min/60s * 6.28rad/rev =
-58.6 rad/s. = The change in velocity.
60s/560rev * 50rev = 5.36 s.
a = (-58.6)/5.36 = -10.9 rad/s^2.
To find the angular acceleration, we need to use the angular velocity equation:
ω = ω₀ + α*t
where:
ω is the final angular velocity
ω₀ is the initial angular velocity
α is the angular acceleration
t is the time interval
We are given the initial and final angular velocities (700 rpm and 140 rpm) and the number of revolutions (50 revolutions). We need to convert the angular velocities from rpm to rad/s.
1 revolution = 2π radians
1 minute = 60 seconds
So, the initial angular velocity ω₀ = (700 rpm) * (2π radians/1 revolution) * (1 minute/60 seconds) = 73.33 rad/s
And the final angular velocity ω = (140 rpm) * (2π radians/1 revolution) * (1 minute/60 seconds) = 14.67 rad/s
Next, we need to find the time interval t. Since we are given the number of revolutions, we can use the formula:
t = (number of revolutions) / (revolutions per minute) * (1 minute/60 seconds)
t = 50 revolutions / 700 rpm * (1 minute/60 seconds) = 0.119 seconds
Now, we can rearrange the angular velocity equation to solve for the angular acceleration α:
α = (ω - ω₀) / t
α = (14.67 rad/s - 73.33 rad/s) / 0.119 seconds
α = -58.66 rad/s / 0.119 seconds
α ≈ -493 rad/s²
Therefore, the angular acceleration through these 50 revolutions is approximately -493 rad/s².