A centrifuge in a medical laboratory rotates at an angular speed of 3,550 rev/min. When switched off, it rotates through 54.0 revolutions before coming to rest. Find the constant angular acceleration (in rad/s2) of the centrifuge.
To find the constant angular acceleration of the centrifuge, we can use the formula:
θ = ω_initial * t + (1/2) * α * t^2
Where:
θ is the total angle rotated (in radians),
ω_initial is the initial angular speed (in rad/s),
t is the time taken (in seconds),
and α is the angular acceleration (in rad/s^2).
We are given:
ω_initial = 3,550 rev/min = (3550 rev/min) * (2π rad/rev) * (1/60 min/s) = 371.94 rad/s
θ = 54 revolutions = (54 rev) * (2π rad/rev) = 340.18 rad
Since the centrifuge comes to rest, its final angular velocity ω_final is 0 rad/s.
Using the formula, we can solve for the angular acceleration α:
θ = ω_initial * t + (1/2) * α * t^2
Rearranging the equation, we get:
0 = ω_initial * t + (1/2) * α * t^2
Since the centrifuge comes to rest, the final time t can be calculated using the formula:
ω_final = ω_initial + α * t
0 = 371.94 rad/s + α * t
Solving for t, we get:
t = -371.94 rad/s / α
Substituting this value of t back into the first equation, we get:
0 = ω_initial * (-371.94 rad/s / α) + (1/2) * α * (-371.94 rad/s / α)^2
Simplifying the equation further, we get:
0 = -371.94 rad/s + (1/2) * 371.94^2 rad^2/s^2 / α
Rearranging the equation, we get:
371.94 rad/s = (1/2) * 371.94^2 rad^2/s^2 / α
Multiplying both sides by 2/371.94^2, we get:
(2/371.94^2) * 371.94 rad/s = α
Simplifying further, we get:
α ≈ 3.98 x 10^-5 rad/s^2
Therefore, the constant angular acceleration of the centrifuge is approximately 3.98 x 10^-5 rad/s^2.
To find the constant angular acceleration of the centrifuge, we can use the equation:
θ = ω_i * t + 1/2 * α * t^2
where θ is the total angle rotated, ω_i is the initial angular speed, t is the time taken, and α is the angular acceleration.
Given values:
ω_i = 3,550 rev/min (initial angular speed)
θ = 54.0 revolutions (total angle rotated)
First, let's convert the initial angular speed from rev/min to rad/s:
1 revolution = 2π radians
ω_i = 3,550 rev/min * (2π rad/1 rev) * (1 min/60 s) = 372.5 rad/s
Now, we can substitute the values into the equation and solve for α:
θ = ω_i * t + 1/2 * α * t^2
Plugging in the values:
54.0 revolutions = (372.5 rad/s) * t + 1/2 * α * t^2
Since the centrifuge comes to rest, its final angular speed is zero. This means that ω_f = 0.
We can use the equation to find the time taken for the centrifuge to come to rest:
ω_f = ω_i + α * t
0 = 372.5 rad/s + α * t_rest
Since the centrifuge rotates through 54.0 revolutions before coming to rest, we can substitute ω_f = 0 and solve for t_rest:
0 = 372.5 rad/s + α * t_rest
Now, we can solve the two equations simultaneously to find the values of α and t_rest.
54.0 revolutions = (372.5 rad/s) * t_rest + 1/2 * α * t_rest^2
0 = 372.5 rad/s + α * t_rest
Solving these equations will give us the constant angular acceleration (α) of the centrifuge.
t = 54rev * 1min/3500rev = 0.0154 min. = 0.93s.
Vo = 3550 rev/min * 6.28rad/rev * 1min/60s = 372 rad/s.
V = Vo + a*t = 0, a = -Vo/t = -372/0.93 = -400 rad/s^2.