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 rad/s and α=0.6 rad/s to complete θ=4 radians. Determine the time taken to complete θ=4 radians.
See previous post: Sat, 4-11-15, 4:23 AM.
To determine the time taken to complete θ = 4 radians, we'll use the given equation θ = ωt + 1/2αt^2, where θ is the angle turned, ω is the initial angular velocity, α is the angular acceleration, and t is the time taken.
Given:
ω = 3 rad/s
α = 0.6 rad/s^2
θ = 4 radians
Substituting the given values into the equation, we have:
4 = (3)t + (1/2)(0.6)t^2
Simplifying the equation:
4 = 3t + 0.3t^2
Rearranging the equation to standard 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.
Calculating the discriminant:
D = b^2 - 4ac
D = (3)^2 - 4(0.3)(-4)
D = 9 + 4.8
D = 13.8
Since the discriminant D is positive, there are two real solutions for t.
Calculating the solutions using the quadratic formula:
t1 = (-3 + √(13.8)) / (2 * 0.3)
t1 ≈ 1.3521 seconds
t2 = (-3 - √(13.8)) / (2 * 0.3)
t2 ≈ -11.0195 seconds
Since time cannot be negative in this context, we discard the negative solution.
Therefore, the time taken to complete θ = 4 radians is approximately 1.3521 seconds.