A motor takes 5 s to go from 30 rad/s to 130 rad/s at constant angular acceleration. The total angle through which its shaft turns during this time is...
A. 400 rad
B. 320 rad
C. 640 rad
D. 800 rad
E. 1600 rad
F. None of the above
V = Vo + a*t = 130 rad/s.
30 + a*5 = 130, a = 20 rad/s^2.
d = Vo*t + 0.5a*t^2.
d = 30*5 + 10*5^2 = 400 rad.
To find the total angle through which the shaft turns, we need to use the kinematic equation for rotational motion:
ω^2 = ω₀^2 + 2αθ
Where:
ω = final angular velocity
ω₀ = initial angular velocity
α = angular acceleration
θ = angle
In this case, we are given:
ω₀ = 30 rad/s
ω = 130 rad/s
α = unknown
θ = unknown
We can solve for α using the equation:
ω = ω₀ + αt
We know that the time, t, is 5 seconds, so plugging in the values:
130 rad/s = 30 rad/s + α * 5 s
Solving for α, we get:
α = (130 rad/s - 30 rad/s) / 5 s
α = 100 rad/s / 5 s
α = 20 rad/s²
Now that we have α, we can plug it into the first equation to solve for θ:
130^2 = 30^2 + 2 * 20 * θ
Simplifying the equation:
16900 = 900 + 40θ
Subtracting 900 from both sides:
16000 = 40θ
Dividing both sides by 40:
θ = 400 rad
Therefore, the total angle through which the shaft turns during this time is 400 rad.
So, the correct answer is A. 400 rad.