The angle of a rotating shaft turns in t seconds is given by theta= ω+1/2at^2
and it is given that ω=3 2
rad/s and α=0.6 rad/s to complete θ=4 radians. Determine the time taken to complete θ=4 radians.
Theta = 4 Rad.
Wo = 3.2? Rad/s
a = 0.6rad/s^2
Theta = Wo*t + 0.5a*t^2 = 4 rads.
3.2t + 0.5*0.6t^2 = 4
0.3t^2 + 3.2t - 4 = 0
Use Quadratic Formula.
t = 1.13 s (Wo = 3.2 rad/s).
t = 0.125 s (Wo = 32 rad/s).
1.191s
Why did the rotating shaft go to therapy?
Because it needed to find its theta-utic angle!
Now, let's solve this question. We're given ω = 32 rad/s and α = 0.6 rad/s. We want to find the time taken to complete θ = 4 radians.
Using the formula θ = ωt + 1/2αt², we can plug in the values we have.
4 = (32)t + 1/2(0.6)t²
Let's solve this equation to find our answer!
To determine the time taken to complete θ=4 radians, we can use the equation provided:θ = ωt + 1/2αt^2
Given:
θ = 4 radians
ω = 3 rad/s
α = 0.6 rad/s^2
Substituting these values into the equation, we have:
4 = (3)t + 1/2(0.6)t^2
Simplifying the equation gives:
4 = 3t + 0.3t^2
Rearranging the equation in quadratic form:
0.3t^2 + 3t - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 0.3, b = 3, and c = -4.
Substituting these values into the formula, we get:
t = (-(3) ± √((3)^2 - 4(0.3)(-4))) / (2(0.3))
Simplifying further:
t = (-3 ± √(9 + 4.8)) / 0.6
t = (-3 ± √13.8) / 0.6
Now, we can calculate the two possible values of t using the positive and negative signs:
t1 = (-3 + √13.8) / 0.6
t2 = (-3 - √13.8) / 0.6
Calculating the values gives:
t1 ≈ 1.74 seconds
t2 ≈ -9.07 seconds
Since time cannot be negative, we discard t2 ≈ -9.07 seconds as the extraneous solution. Therefore, the time taken to complete θ=4 radians is approximately 1.74 seconds.