After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of +1.88 rad/s. During this time, the wheel has an angular acceleration of `-5.04rad//s^(2)`. Determine the angular displacement of the wheel.

Options

A. `+440` rad
B. `+252` rad
C. `+271` rad
D. `+188` rad

Answer

To determine the angular displacement of the roulette wheel, we can use the equations of rotational motion. Here’s the step-by-step solution:

### Step 1: Identify the known variables
- Final angular velocity (\( \omega_f \)) = +1.88 rad/s
- Angular acceleration (\( \alpha \)) = -5.04 rad/s²
- Time (\( t \)) = 10.0 s

### Step 2: Use the angular velocity equation to find the initial angular velocity
We can use the equation:
\[ \omega_f = \omega_i + \alpha t \]
Rearranging this to solve for the initial angular velocity (\( \omega_i \)):
\[ \omega_i = \omega_f - \alpha t \]
Substituting the known values:
\[ \omega_i = 1.88 \, \text{rad/s} - (-5.04 \, \text{rad/s}^2 \times 10.0 \, \text{s}) \]
Calculating:
\[ \omega_i = 1.88 \, \text{rad/s} + 50.4 \, \text{rad/s} = 52.28 \, \text{rad/s} \]

### Step 3: Use the angular displacement formula
The angular displacement (\( \theta \)) can be calculated using the formula:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the values we have:
\[ \theta = (52.28 \, \text{rad/s} \times 10.0 \, \text{s}) + \frac{1}{2} (-5.04 \, \text{rad/s}^2) (10.0 \, \text{s})^2 \]

### Step 4: Calculate each term
Calculating the first term:
\[ 52.28 \, \text{rad/s} \times 10.0 \, \text{s} = 522.8 \, \text{rad} \]
Calculating the second term:
\[ \frac{1}{2} \times (-5.04 \, \text{rad/s}^2) \times 100 \, \text{s}^2 = -252 \, \text{rad} \]

### Step 5: Combine the results to find angular displacement
Now, we combine both terms:
\[ \theta = 522.8 \, \text{rad} - 252 \, \text{rad} = 270.8 \, \text{rad} \]

### Step 6: Round to the appropriate significant figures
The final angular displacement is approximately:
\[ \theta \approx 271 \, \text{rad} \]

### Final Answer
The angular displacement of the wheel is approximately **271 radians**.

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