You are travelling forward in a car at 20 m/s. Your car tires have a radius of 33 cm. You slam on the brakes, which slows the tires at an angular acceleration of 40 rad/s^2.
How long until the tires are no longer spinning (although the car may still be sliding forward) in s?
To solve this problem, we need to first determine the angular velocity of the tires when the car starts to brake. We can use the formula:
ω = ω0 + αt
Where:
ω = angular velocity (final)
ω0 = initial angular velocity (when the car starts to brake)
α = angular acceleration
t = time
Since the car tires are initially spinning at a constant speed, the initial angular velocity is simply the linear velocity divided by the tire radius:
ω0 = v / r
Plugging in the values:
ω0 = 20 m/s / 0.33 m = 60.606 rad/s
Next, we'll use the formula for angular velocity to calculate the time it takes for the tires to stop spinning:
0 = ω0 + αt
Rearranging the equation:
t = (-ω0) / α
Plugging in the values:
t = (-60.606 rad/s) / (40 rad/s^2)
t = -1.515 s
Since time cannot be negative, we'll take the positive value:
t = 1.515 s
Therefore, it takes approximately 1.515 seconds for the car tires to stop spinning.