A wheel has an angular speed of 85 rev/s when it experiences a constant angular acceleration of 100 rev/s2 which causes it to spin FASTER. During this time the wheel completes 26 rev. Determine how long the wheel was experiencing this angular acceleration.
100 t^2 + 85 t = 26 ... 100 t^2 + 85 t - 26 = 0
use the quadratic formula to find t
To determine how long the wheel was experiencing the angular acceleration, we can use the following equation:
ω^2 = ω0^2 + 2αθ
Where:
ω = Final angular speed (in rad/s)
ω0 = Initial angular speed (in rad/s)
α = Angular acceleration (in rad/s^2)
θ = Angular displacement (in radians)
First, we need to convert the given values of angular speeds and acceleration from rev/s to rad/s:
Convert the final angular speed:
ω = 85 rev/s * 2π rad/rev = 170π rad/s
Convert the initial angular speed:
ω0 = 0 rev/s * 2π rad/rev = 0 rad/s
Convert the angular acceleration:
α = 100 rev/s^2 * 2π rad/rev = 200π rad/s^2
Now we can rearrange the equation to solve for θ:
θ = (ω^2 - ω0^2) / (2α)
θ = (170π rad/s)^2 / (2 * 200π rad/s^2)
θ = 28900π^2 / 400π
θ = 28900 / 400
θ = 72.25 rad
Now we can determine the time by using the equation:
θ = ω0 * t + 0.5 * α * t^2
Rearranging the equation to solve for time:
t = (-ω0 ± √(ω0^2 + 2αθ)) / α
We can ignore the negative solution since time cannot be negative in this case.
t = (√(ω0^2 + 2αθ)) / α
Substituting the values:
t = (√((0 rad/s)^2 + 2 * 200π rad/s^2 * 72.25 rad)) / (200π rad/s^2)
t = (√(0 + 14450π)) / (200π rad/s^2)
t = (√(14450π)) / (200π rad/s^2)
t ≈ (√(14450)) / 200 ≈ 5.38 seconds
Therefore, the wheel was experiencing this angular acceleration for approximately 5.38 seconds.