Consider a car that has come to a complete stop at a red light. When the light turns green, the car begins to accelerate forward with a constant acceleration a=1.4 m/s2.

The tires on the car, which have radius r=0.33 m, roll without slipping while the car accelerates.

What will be the angular velocity of the car's wheels after they have completed at total of five rotations?

(a)

9.38 rad/s

(b)

6.51 rad/s

(c)

16.33 rad/s

To determine the angular velocity of the car's wheels after five rotations, we can follow these steps:

1. **Convert Rotations to Distance**: First, calculate the linear distance the car has traveled after five rotations of the wheels.

   \[
   \text{Circumference of the wheel} = 2 \pi r
   \]
   where \( r = 0.33 \, \text{m} \).

   So, for five rotations:

   \[
   \text{Distance traveled} = 5 \times 2 \pi \times 0.33 \approx 10.37 \, \text{m}
   \]

2. **Use Kinematics to Find Linear Velocity**: Given the car starts from rest and accelerates with a constant acceleration \( a = 1.4 \, \text{m/s}^2 \), we can use the following kinematic equation:

   \[
   v^2 = u^2 + 2as
   \]
   where:
   - \( v \) is the final linear velocity,
   - \( u = 0 \) (initial velocity),
   - \( a = 1.4 \, \text{m/s}^2 \),
   - \( s = 10.37 \, \text{m} \) (distance traveled).

   Plugging in the values:

   \[
   v^2 = 0 + 2 \times 1.4 \times 10.37
   \]
   \[
   v^2 = 29.036
   \]
   \[
   v \approx 5.39 \, \text{m/s}
   \]

3. **Relate Linear Velocity to Angular Velocity**: Since the wheels roll without slipping, we can relate the linear velocity \( v \) of the car to the angular velocity \( \omega \) of the wheels using:

   \[
   v = r \omega
   \]
   Solving for \( \omega \):

   \[
   \omega = \frac{v}{r} = \frac{5.39}{0.33} \approx 16.33 \, \text{rad/s}
   \]

So, the correct answer is:

**(c) 16.33 rad/s**.

---

Would you like a further breakdown, or do you have any questions?

Here are some related questions to expand on this concept:

1. How would the solution change if the acceleration was doubled?
2. What would be the impact if the car started with an initial velocity instead of from rest?
3. How would frictional forces affect the rolling motion if it were no longer rolling without slipping?
4. What is the relationship between linear and angular acceleration in rolling motion?
5. How would the result differ if the car traveled over a distance of 15 meters instead?

**Tip**: When dealing with rolling motion, always check if there's slipping. Rolling without slipping creates a direct relationship between linear and angular quantities.

Consider a car that has come to a complete stop at a red light. When the light turns green, the car begins to accelerate forward with a constant acceleration a = 1.4 m/s². The tires on the car, which have radius r = 0.33 m, roll without slipping while the car accelerates. What will be the angular velocity of the car's wheels after they have completed a total of five rotations?

(a) 9.38 rad/s
(b) 6.51 rad/s
(c) 16.33 rad/s

Learn how to calculate the angular velocity of a car's wheels given linear acceleration and radius after completing five rotations. This physics problem uses kinematics, rotational motion, and angular velocity concepts, suitable for students in Grades 11-12.

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