A typical ten-pound car wheel has a moment of inertia of about 0.35  kg⋅m2
k
g
⋅
m
2
. The wheel rotates about the axle at a constant angular speed making 70.0 full revolutions in a time interval of 7.00  s
s
.

It seems you're describing a scenario in which a car wheel rotates with a given moment of inertia and completes a certain number of revolutions within a specified time. Let’s break this down:

### Given Data:
- Moment of inertia (\( I \)) = 0.35 kg·m²
- Number of revolutions = 70.0 revolutions
- Time interval = 7.00 s

First, let's calculate the angular velocity of the wheel.

### Step 1: Angular velocity (\( \omega \))
The angular velocity (\( \omega \)) is the rate of change of angular displacement with respect to time. Since the wheel makes 70 full revolutions in 7.00 seconds, we can calculate the number of radians per second.

1 full revolution = \( 2\pi \) radians.

So, total angular displacement for 70 revolutions is:
\[
\theta = 70 \times 2\pi = 140\pi \text{ radians}
\]

Now, the angular velocity \( \omega \) is the total angular displacement divided by the time:
\[
\omega = \frac{\theta}{t} = \frac{140\pi}{7.00} = 20\pi \text{ radians/second}
\]

Thus, the angular velocity of the wheel is:
\[
\omega \approx 62.83 \text{ radians/second}
\]

### Step 2: Kinetic Energy (Rotational)
The rotational kinetic energy (\( K_E \)) of the wheel is given by the equation:
\[
K_E = \frac{1}{2} I \omega^2
\]

Substituting the values:
\[
K_E = \frac{1}{2} \times 0.35 \times (62.83)^2
\]
\[
K_E \approx \frac{1}{2} \times 0.35 \times 3945.5
\]
\[
K_E \approx 691.4 \text{ Joules}
\]

### Conclusion:
The angular velocity of the wheel is approximately \( 62.83 \, \text{radians/second} \), and the rotational kinetic energy is approximately \( 691.4 \, \text{Joules} \).

Would you like to explore any more aspects of this problem? Here are some related questions you could consider:
1. How does the moment of inertia affect the angular velocity for a rotating object?
2. What would happen if the wheel were rotating faster or slower? How would it affect the energy?
3. How can you calculate the angular acceleration if the wheel started from rest?
4. How does the radius of the wheel affect the moment of inertia and rotational energy?
5. What is the relationship between linear velocity and angular velocity?

Tip: **Understanding the rotational kinetic energy equation helps in understanding how much energy is needed to maintain a constant rotational speed in an object.**

A typical ten-pound car wheel has a moment of inertia of about 0.35 kg⋅m². The wheel rotates about the axle at a constant angular speed making 70.0 full revolutions in a time interval of 7.00 s.

Explore the physics behind a rotating car wheel with a moment of inertia of 0.35 kg⋅m², completing 70 revolutions in 7 seconds. Learn how to calculate angular velocity (62.83 rad/s) and rotational kinetic energy (691.4 Joules) with detailed steps and formulas. Ideal for high school students studying rotational motion.

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