A tangential for of 8.0 N acts on a wheel which com plates its first revolution from rest in 20 s s the force acts 0.30 m from the center of the wheel. Calculate a. The angular acceleration b. The moment of inertia of the wheel

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**Step 1. Determine the Angular Acceleration**

The wheel starts from rest and completes one full revolution in $$ t = 20~\text{s} $$. One full revolution corresponds to an angular displacement of  
$$
\theta = 2\pi~\text{radians}.
$$

For constant angular acceleration, the angular displacement is given by
$$
\theta = \frac{1}{2}\alpha t^2.
$$

Solving for the angular acceleration $$ \alpha $$,
$$
\alpha = \frac{2\theta}{t^2}.
$$

Substitute the known values,
$$
\alpha = \frac{2(2\pi)}{(20)^2} = \frac{4\pi}{400} = \frac{\pi}{100}~\text{rad/s}^2.
$$

Thus, the angular acceleration is  
$$
\boxed{\alpha \approx 0.0314~\text{rad/s}^2}.
$$

---

**Step 2. Calculate the Moment of Inertia**

A tangential force $$ F = 8.0~\text{N} $$ acts at a distance $$ r = 0.30~\text{m} $$ from the center. The torque $$ \tau $$ produced by this force is given by
$$
\tau = F \cdot r.
$$

Substitute the values,
$$
\tau = 8.0~\text{N} \times 0.30~\text{m} = 2.4~\text{N}\cdot\text{m}.
$$

The relation between torque and angular acceleration is
$$
\tau = I \alpha,
$$
where $$ I $$ is the moment of inertia. Solving for $$ I $$,
$$
I = \frac{\tau}{\alpha}.
$$

Now substitute the known values,
$$
I = \frac{2.4~\text{N}\cdot\text{m}}{\frac{\pi}{100}~\text{rad/s}^2} = \frac{2.4 \times 100}{\pi} = \frac{240}{\pi}~\text{kg}\cdot\text{m}^2.
$$

Thus, the moment of inertia is approximately
$$
\boxed{I \approx 76.4~\text{kg}\cdot\text{m}^2}.
$$
**a. Angular Acceleration:** The wheel accelerates at approximately $$ 0.0314~\text{rad/s}^2 $$. **b. Moment of Inertia:** The wheel has a moment of inertia of approximately $$ 76.4~\text{kg}\cdot\text{m}^2 $$.

A tangential for of 8.0 N acts on a wheel which com plates its first revolution from rest in 20 s s the force acts 0.30 m from the center of the wheel. Calculate a. The angular acceleration b. The moment of inertia of the wheel

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