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.
It takes 1.515 seconds for the tires to stop spinning. Through how many radians of rotation does each tire go from the time you slam on the brakes until the tires stop spinning?
To find the number of radians of rotation, we need to calculate the angular displacement of each tire.
We can start by finding the initial angular velocity (ω) of the tires. The linear velocity (v) of the car is given as 20 m/s, and the radius (r) of the tires is 33 cm (or 0.33 m).
The linear velocity of a point on the tire is given by:
v = r * ω
Rearranging the equation, we can solve for ω:
ω = v / r
ω = 20 m/s / 0.33 m
ω ≈ 60.606 rad/s
Now, we can use the following equation to find the angular displacement (θ) using the initial angular velocity (ω), angular acceleration (α), and time (t):
θ = ω*t + (1/2)*α*t^2
Plugging in the values, we have:
θ = (60.606 rad/s)(1.515 s) + (1/2)(40 rad/s^2)(1.515 s)^2
θ ≈ 91.817 rad
Therefore, each tire goes through approximately 91.817 radians of rotation from the time the brakes are slammed until the tires stop spinning.