A basketball player is balancing a spinning basketball on the tip of his finger. The angular velocity of the ball slows down from 18.5 to 14.1 rad/s. During the slow-down, the angular displacement is 85.1 rad. Determine the time it takes for the ball to slow down.

Options

A. 4.50 s
B. 5.22 s
C. 3.11 s
D. 4.76 s

Answer

To determine the time it takes for the basketball to slow down, we can use the equations of angular motion. Here’s a step-by-step solution:

### Step 1: Identify the known values
- Initial angular velocity, \( \omega_i = 18.5 \, \text{rad/s} \)
- Final angular velocity, \( \omega_f = 14.1 \, \text{rad/s} \)
- Angular displacement, \( \theta = 85.1 \, \text{rad} \)

### Step 2: Use the angular motion equation
We can use the equation that relates initial and final angular velocities, angular displacement, and angular acceleration:
\[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]
Since the basketball is slowing down, the angular acceleration \( \alpha \) will be negative. Therefore, we can rewrite the equation as:
\[ \omega_f^2 = \omega_i^2 - 2|\alpha|\theta \]

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ (14.1)^2 = (18.5)^2 - 2|\alpha|(85.1) \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 198.81 = 342.25 - 2|\alpha|(85.1) \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 2|\alpha|(85.1) = 342.25 - 198.81 \]
\[ 2|\alpha|(85.1) = 143.44 \]

### Step 6: Solve for \( |\alpha| \)
Now, divide both sides by \( 2 \times 85.1 \):
\[ |\alpha| = \frac{143.44}{2 \times 85.1} \]
Calculating this gives:
\[ |\alpha| = \frac{143.44}{170.2} \approx 0.842 \, \text{rad/s}^2 \]

### Step 7: Use the angular acceleration to find time
Now, we can use the angular acceleration to find the time taken to slow down using the equation:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the known values:
\[ 14.1 = 18.5 - 0.842 t \]

### Step 8: Rearrange to solve for \( t \)
Rearranging gives:
\[ 0.842 t = 18.5 - 14.1 \]
\[ 0.842 t = 4.4 \]
\[ t = \frac{4.4}{0.842} \approx 5.22 \, \text{s} \]

### Final Answer
The time it takes for the basketball to slow down is approximately **5.22 seconds**.

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