A wheel of radius 0.43 m spins at a rate such that a point on the rim of the wheel has an acceleration of 43 m/s^2. How many rotations does the wheel make in a 12 minute drive?
To find the number of rotations the wheel makes in a 12-minute drive, we need to convert the time into the unit of seconds.
1 minute is equal to 60 seconds, so 12 minutes is equal to 12 * 60 = 720 seconds.
We know that the acceleration of a point on the rim of the wheel is 43 m/s^2, and we can use this information to find the angular acceleration of the wheel.
The linear acceleration (a) is related to the angular acceleration (α) by the formula a = α * r, where r is the radius of the wheel.
In this case, the linear acceleration is 43 m/s^2 and the radius of the wheel is 0.43 m. So, we have:
43 = α * 0.43
To find the angular acceleration (α), we divide both sides of the equation by 0.43:
α = 43 / 0.43 = 100 rad/s^2
Now, we can use the angular acceleration to find the angular velocity and the number of rotations.
The formula for finding angular velocity (ω) is ω = α * t, where t is the time.
In this case, the time is 720 seconds and the angular acceleration is 100 rad/s^2. So, we have:
ω = 100 * 720 = 72,000 rad/s
The number of rotations (N) can be found using the formula N = ω / (2π), where ω is the angular velocity and π is the mathematical constant pi.
In this case, the angular velocity is 72,000 rad/s. So, we have:
N = 72,000 / (2π) ≈ 11,461 rotations
Therefore, the wheel makes approximately 11,461 rotations in a 12-minute drive.