The angular speed of a rotor in a centrifuge increases from 428 to 1400 rad/s in a time of 6.26 s. (a) Obtain the angle through which the rotor turns. (b) What is the magnitude of the angular acceleration?
i figured it out,
[v(initial)*time]+[[acceleration*time^2]/2]
d omega / dt = alpha = (1400-428)/6.26
= 155 rad/s^2 which is angular acceleration part b
theta = 428 t + (1/2)(155)t^2
but t is 6.26
so
theta = 5722 radians which is part a
To solve this problem, we can use the equations of angular motion. Let's go step-by-step:
(a) To find the angle through which the rotor turns, we can use the formula:
θ = ω1 * t + (1/2) * α * t^2
where θ is the angle through which the rotor turns, ω1 is the initial angular speed, t is the time, and α is the angular acceleration.
Given:
ω1 = 428 rad/s
t = 6.26 s
To find α, we need to calculate the change in angular speed:
Δω = ω2 - ω1
where ω2 is the final angular speed.
Given:
ω2 = 1400 rad/s
Δω = 1400 rad/s - 428 rad/s
Δω = 972 rad/s
Now, rearranging the equation, we have:
θ = ω1 * t + (1/2) * α * t^2
θ = 428 rad/s * 6.26 s + (1/2) * α * (6.26 s)^2
To find θ, we need to calculate α. Rearranging the equation, we have:
α = (2 * (θ - ω1 * t)) / (t^2)
Now we can substitute the known values:
α = (2 * (θ - 428 rad/s * 6.26 s)) / (6.26 s)^2
Simplifying the equation, we get:
α = (2 * (θ - 2682.8)) / 39.1876
(b) To find the magnitude of the angular acceleration (α), we can use the equation above. Rearranging the equation, we have:
α = (2 * (θ - 2682.8)) / 39.1876
Now we need to substitute the values:
α = (2 * (θ - 2682.8)) / 39.1876
α = (2 * (θ - 2682.8)) / 39.1876
Now you have two equations with two unknowns (θ and α). You can solve them simultaneously to find the values of θ and α.
To solve this problem, we'll use the kinematic equations for rotational motion. Let's go step-by-step to find the answers.
(a) To obtain the angle through which the rotor turns, we can use the formula:
θ = ω1*t + (1/2)*α*t^2
Where:
θ is the angle through which the rotor turns,
ω1 is the initial angular speed,
t is the time taken,
α is the angular acceleration.
Given:
ω1 = 428 rad/s
t = 6.26 s
To find α, we need the final angular speed. We know that:
ω2 = ω1 + α*t
Rearranging the equation, we have:
α = (ω2 - ω1) / t
Given:
ω2 = 1400 rad/s
t = 6.26 s
Now, substituting the values, we can find α:
α = (1400 - 428) / 6.26
Next, we'll use the above value of α along with the given values of ω1 and t to find θ:
θ = ω1*t + (1/2)*α*t^2
Substituting the values, we get:
θ = (428)*(6.26) + (1/2)*(α)*(6.26)^2
Simplifying the equation gives us the value of θ.
(b) To find the magnitude of the angular acceleration, we can simply use the value of α that we found earlier.
Now, follow these steps to calculate the answer:
1. Calculate α using the formula: α = (ω2 - ω1) / t
2. Calculate θ using the formula: θ = ω1*t + (1/2)*α*t^2
3. Calculate the magnitude of the angular acceleration, which is simply the value of α.
Substituting the given values into the formulas will give you the desired answers.