A person is riding a bicycle, and its wheels have an angular velocity of +20.0 rad/s Then, the brakes are applied, and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +15.92 rev. How much time does it take for the bike to come to rest?

Options

A. 5.00 s
B. 10.0 s
C. 3.18 s
D. 6.36 s

Answer

To solve the problem step by step, we will follow the physics principles related to angular motion.

### Step 1: Identify the given values
- Initial angular velocity (\( \omega_i \)) = +20.0 rad/s
- Final angular velocity (\( \omega_f \)) = 0 rad/s (since the bike comes to rest)
- Angular displacement (\( \theta \)) = +15.92 revolutions

### Step 2: Convert angular displacement from revolutions to radians
To convert revolutions to radians, we use the conversion factor \( 2\pi \) radians per revolution:
\[ \theta = 15.92 \, \text{rev} \times 2\pi \, \text{rad/rev} = 15.92 \times 2\pi \, \text{rad} \]
Calculating this gives:
\[ \theta \approx 15.92 \times 6.2832 \approx 100.0 \, \text{rad} \]

### Step 3: Use the angular kinematic equation
We can use the following angular kinematic equation:
\[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]
Since the bike is coming to rest, \( \omega_f = 0 \):
\[ 0 = (20.0)^2 + 2\alpha(100.0) \]
This simplifies to:
\[ 0 = 400 + 200\alpha \]
Rearranging gives:
\[ 200\alpha = -400 \implies \alpha = -2.0 \, \text{rad/s}^2 \]
The negative sign indicates that it is a deceleration (retardation).

### Step 4: Calculate the time taken to come to rest
We can use another angular kinematic equation:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the known values:
\[ 0 = 20.0 + (-2.0)t \]
Rearranging gives:
\[ 2.0t = 20.0 \implies t = \frac{20.0}{2.0} = 10.0 \, \text{s} \]

### Final Answer
The time taken for the bike to come to rest is **10 seconds**.

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