A Ferris wheel with radius 9.8 m rotates at a constant rate, completing one revolution every 36.6 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 of the passenger's acceleration at that time.
_____magnitude
direction______° below direction of travel
Well, this Ferris wheel seems to be having a "wheel-y" good time! Let's calculate the magnitude and direction of the passenger's acceleration.
First, we need to find the angular acceleration of the Ferris wheel. We know that the time it takes to complete one revolution is 36.6 seconds, and since there are 2π radians in a full revolution, we can find the angular velocity:
ω = 2π / 36.6 s = 0.1724 rad/s
Now, we can use the formula for angular acceleration:
α = (ωf - ωi) / t
Substituting the values:
0.227 rad/s^2 = (0 - 0.1724 rad/s) / t
Solving for t:
t = (0.1724 rad/s) / 0.227 rad/s^2 = 0.7592 s
So, the time it takes for the passenger to reach the top is 0.7592 seconds.
Now, let's find the linear acceleration at the top of the wheel. We know that linear acceleration is given by:
a = r * α
Substituting the values:
a = 9.8 m * 0.227 rad/s^2 = 2.2296 m/s^2
Therefore, the magnitude of the passenger's acceleration is 2.2296 m/s^2.
Lastly, the direction of the acceleration is "___° below the direction of travel." Since the passenger is at the top of the wheel, the direction of travel is downward. So, the direction of the acceleration is 180° below the direction of travel.
I hope this ferris-wheelful answer brings a little amusement to your day!
To find the magnitude and direction of the passenger's acceleration at the top of the Ferris wheel, we can use the following steps:
Step 1: Determine the angular velocity and angular acceleration of the Ferris wheel.
Given that the Ferris wheel completes one revolution every 36.6 s, we can calculate its angular velocity using the formula:
Angular velocity (ω) = (2π) / time period = (2π) / 36.6 s.
Since the Ferris wheel is decelerating, its angular acceleration will be negative. Given that the angular acceleration is -0.227 rad/s^2, we have:
Angular acceleration (α) = -0.227 rad/s^2.
Step 2: Calculate the linear speed and linear acceleration of the passenger.
The linear speed (v) of a point on the Ferris wheel can be calculated using the formula:
v = r * ω,
where r is the radius of the Ferris wheel.
Substituting the given values, we have:
v = 9.8 m * (2π / 36.6 s).
The linear acceleration (a) of a point on a rotating object can be calculated using the formula:
a = r * α.
Substituting the given values, we have:
a = 9.8 m * -0.227 rad/s^2.
Step 3: Determine the magnitude and direction of the passenger's acceleration.
The magnitude of the passenger's acceleration can be found using the formula:
Magnitude of acceleration (|a|) = √(a^2 + v^2).
Substituting the calculated values, we have:
|a| = √((-9.8 m * 0.227 rad/s^2)^2 + (9.8 m * (2π / 36.6 s))^2).
The direction of the passenger's acceleration can be calculated using the formula:
θ = arctan(v / a).
Substituting the calculated values, we have:
θ = arctan((9.8 m * (2π / 36.6 s)) / (-9.8 m * 0.227 rad/s^2)).
Calculate these values to find the magnitude and direction of the passenger's acceleration.