Suppose that a car wheel is spinning with angular velocity of wo. The car is then turned off and the wheel start to slow down because of friction and angular acceleration is constant. The wheel comes to rest after it has made X revolutions. What is the magnitude of the angular acceleration?
average velocity is ... wo / 2
time to stop is ... rev / vel = 2 X / wo
acc = vel / time = wo^2 / (2 X)
To determine the magnitude of the angular acceleration, we can use the equation relating angular velocity, angular acceleration, and the number of revolutions.
The equation is:
ω² = ω₀² + 2αθ
Where:
ω is the final angular velocity (0 in this case because the wheel comes to rest)
ω₀ is the initial angular velocity (given as wo)
α is the angular acceleration (what we are trying to find)
θ is the angle covered (in this case, X revolutions or 2πX)
Plugging in the given values into the equation and solving for α, we get:
0² = wo² + 2α(2πX)
Simplifying further:
0 = wo² + 4παX
Solving for α:
α = -(wo²) / (4πX)
Therefore, the magnitude of the angular acceleration is given by:
α = -(wo²) / (4πX)