A 3.45-kg centrifuge takes 91 s to spin up from rest to its final angular speed with constant angular acceleration. A point located 8.3 cm from the axis of rotation of the centrifuge moves with a speed of 177 m/s when the centrifuge is at full speed. What is the angular acceleration (in rad/s2) of the centrifuge as it spins up?
To find the angular acceleration of the centrifuge, we can use the following formula:
ω = ω0 + αt
Where:
ω - Final angular speed (in rad/s)
ω0 - Initial angular speed (in rad/s)
α - Angular acceleration (in rad/s^2)
t - Time taken to spin up (in seconds)
Given that the centrifuge takes 91 seconds to spin up from rest to its final angular speed, we can replace the values we know into the equation:
ω = ω0 + αt
177 m/s = ω0 + α * 91 s
Next, we need to find the initial angular speed, ω0, using the formula:
v = ω × r
Where:
v - Linear speed (in m/s)
ω - Angular speed (in rad/s)
r - Radius (in meters)
Given that the point located 8.3 cm from the axis of rotation moves with a speed of 177 m/s, we can convert 8.3 cm to meters and apply the formula:
v = ω × r
177 m/s = ω × 0.083 m
Solving for ω, we find:
ω = 177 m/s ÷ 0.083 m
Now, we have the value of ω0, and we can substitute it back into the first equation:
177 m/s ÷ 0.083 m = ω0 + α * 91 s
Simplifying the equation, we have:
2127.71 rad/s = ω0 + 91 α
To find α, isolate α on one side of the equation:
91 α = 2127.71 rad/s - ω0
And finally, divide both sides of the equation by 91 to solve for α:
α = (2127.71 rad/s - ω0) ÷ 91
Plug in the value of ω0, calculate the expression, and you will find the angular acceleration of the centrifuge in rad/s^2.