A Ferris wheel with radius 11.0 m rotates at a constant rate, completing one revolution every 33.9 s. Suppose the Ferris wheel begins to decelerate at the rate of 0.227 rad/s2 when a passenger is at the top of the wheel. Find the magnitude and direction (in degrees below direction of travel) of the passenger's acceleration at that time.

To find the magnitude and direction of the passenger's acceleration at the top of the wheel, we can follow these steps:

Step 1: Find the angular velocity (ω) of the Ferris wheel in radians per second.
The Ferris wheel completes one revolution in 33.9 seconds, so the time for one revolution is equal to the period (T) of the Ferris wheel.
T = 33.9 s

The angular velocity (ω) is defined as the angle covered per unit time.
ω = 2π / T
= 2π / 33.9 s

Step 2: Calculate the initial angular velocity (ω0) of the Ferris wheel at the top.
Since the passenger is at the top of the wheel, the initial angular velocity (ω0) is equal to the angular velocity (ω) at that time.
ω0 = ω

Step 3: Calculate the final angular velocity (ωf) of the Ferris wheel at the top when it decelerates at a rate of 0.227 rad/s^2.
The final angular velocity (ωf) can be calculated using the following equation:
ωf = ω0 - α * t
where α is the angular acceleration and t is the time.

Given α = -0.227 rad/s^2 and t = 33.9 s, we can substitute these values into the equation:
ωf = ω0 - α * t
ωf = ω - (-0.227 rad/s^2) * 33.9 s

Step 4: Calculate the acceleration (a) of the passenger at the top of the Ferris wheel.
The acceleration (a) can be calculated using the following equation:
a = r * α
where r is the radius of the Ferris wheel.

Given r = 11.0 m and α = -0.227 rad/s^2, we can substitute these values into the equation:
a = (11.0 m) * (-0.227 rad/s^2)

Step 5: Convert the magnitude of acceleration (|a|) from m/s^2 to degrees below the direction of travel.
To convert the magnitude of acceleration (|a|) from m/s^2 to degrees below the direction of travel, we need to use the fact that there are 360 degrees in a circle and 2π radians in a circle.

Let's calculate |a| and convert it to degrees:
|a| = (11.0 m) * (0.227 rad/s^2) * (180° / π rad)
≈ 44.0°

Therefore, the magnitude of the passenger's acceleration at the top of the Ferris wheel is approximately 44.0°.

Step 6: Determine the direction of the passenger's acceleration.
Since the acceleration is negative (-0.227 rad/s^2) and the passenger is at the top of the Ferris wheel, the acceleration is directed downward. This corresponds to 180° below the direction of travel.

Combining the magnitude and direction, the passenger's acceleration at the top of the Ferris wheel is approximately 44.0° below the direction of travel and directed downward.