A car is moving at a speed of `72 kmh^(-1)`. The diameter of its wheel is 0.50m. If the wheels are stopped in 20 rotations by applying brakes, calculate the angular retardation produced by the brakes.

Answer

To solve the problem step by step, we will follow the given information and apply the relevant physics formulas.

### Step 1: Convert the speed from km/h to m/s
The initial speed of the car is given as 72 km/h. We need to convert this to meters per second (m/s).

\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 72 \times \frac{1000}{3600} = 20 \text{ m/s} \]

### Step 2: Calculate the radius of the wheel
The diameter of the wheel is given as 0.50 m, so the radius \( r \) is:

\[ r = \frac{\text{Diameter}}{2} = \frac{0.50 \text{ m}}{2} = 0.25 \text{ m} \]

### Step 3: Calculate the initial angular velocity
The initial linear velocity \( V_0 \) is related to the angular velocity \( \omega_0 \) by the formula:

\[ \omega_0 = \frac{V_0}{r} \]

Substituting the values:

\[ \omega_0 = \frac{20 \text{ m/s}}{0.25 \text{ m}} = 80 \text{ rad/s} \]

### Step 4: Calculate the angular displacement
The angular displacement \( \theta \) in radians for 20 rotations can be calculated as:

\[ \theta = 2\pi \times \text{Number of rotations} = 2\pi \times 20 = 40\pi \text{ rad} \]

### Step 5: Use the third equation of motion for rotational motion
We can use the equation:

\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]

Since the wheels are stopped, the final angular velocity \( \omega \) is 0. Thus, we can rearrange the equation to solve for angular retardation \( \alpha \):

\[ 0 = \omega_0^2 + 2\alpha\theta \]

This simplifies to:

\[ \alpha = -\frac{\omega_0^2}{2\theta} \]

### Step 6: Substitute the values to find angular retardation
Substituting the values we have:

\[ \alpha = -\frac{(80 \text{ rad/s})^2}{2 \times 40\pi \text{ rad}} = -\frac{6400}{80\pi} = -\frac{80}{\pi} \approx -25.5 \text{ rad/s}^2 \]

### Final Answer:
The angular retardation produced by the brakes is approximately \( -25.5 \text{ rad/s}^2 \).

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