A wheel rotates for 5 seconds with a constant angular acceleration and describes during this time 100 radians.It then rotates with a constant angular velocity and during the next five seconds describes 80 radians.Find the initial angular velocity and the angular acceleration

To solve this problem, we need to break it into two parts:

1. **The first part (t = 0 to t = 5 seconds)**, where the wheel has constant angular acceleration.
2. **The second part (t = 5 to t = 10 seconds)**, where the wheel has constant angular velocity.

Let’s define the variables:

- \( \theta_1 = 100 \, \text{radians} \): The angle described in the first 5 seconds.
- \( \theta_2 = 80 \, \text{radians} \): The angle described in the next 5 seconds.
- \( \alpha \): The angular acceleration during the first 5 seconds (to be determined).
- \( \omega_0 \): The initial angular velocity (to be determined).
- \( \omega_f \): The final angular velocity at the end of the first 5 seconds, which is also the constant angular velocity during the second part.

### First part (constant angular acceleration):
We know the relationship between angular displacement, initial angular velocity, angular acceleration, and time under constant acceleration:

\[
\theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2
\]

For the first part of the motion (with \( t = 5 \) seconds and \( \theta_1 = 100 \) radians), this equation becomes:

\[
100 = \omega_0 (5) + \frac{1}{2} \alpha (5)^2
\]
\[
100 = 5 \omega_0 + \frac{25}{2} \alpha
\]
\[
100 = 5 \omega_0 + 12.5 \alpha \quad \text{(Equation 1)}
\]

Also, the angular velocity at the end of the first 5 seconds (which becomes the constant angular velocity for the second part) is given by:

\[
\omega_f = \omega_0 + \alpha t
\]

At \( t = 5 \) seconds:

\[
\omega_f = \omega_0 + 5 \alpha
\quad \text{(Equation 2)}
\]

### Second part (constant angular velocity):
During the second part, the wheel rotates with constant angular velocity \( \omega_f \), and the displacement is given by:

\[
\theta_2 = \omega_f \cdot t
\]

For \( \theta_2 = 80 \) radians and \( t = 5 \) seconds:

\[
80 = \omega_f \cdot 5
\]
\[
\omega_f = \frac{80}{5} = 16 \, \text{rad/s} \quad \text{(Equation 3)}
\]

### Solving the system of equations:

From **Equation 2**, we know that:

\[
\omega_f = \omega_0 + 5 \alpha = 16
\]
\[
\omega_0 + 5 \alpha = 16 \quad \text{(Equation 4)}
\]

Now we have two equations:
1. \( 100 = 5 \omega_0 + 12.5 \alpha \)
2. \( \omega_0 + 5 \alpha = 16 \)

Let’s solve this system. First, solve Equation 4 for \( \omega_0 \):

\[
\omega_0 = 16 - 5 \alpha
\]

Substitute this into Equation 1:

\[
100 = 5(16 - 5 \alpha) + 12.5 \alpha
\]
\[
100 = 80 - 25 \alpha + 12.5 \alpha
\]
\[
100 = 80 - 12.5 \alpha
\]
\[
20 = 12.5 \alpha
\]
\[
\alpha = \frac{20}{12.5} = 1.6 \, \text{rad/s}^2
\]

Now substitute \( \alpha = 1.6 \) into Equation 4 to find \( \omega_0 \):

\[
\omega_0 = 16 - 5(1.6) = 16 - 8 = 8 \, \text{rad/s}
\]

### Final Answer:
- The initial angular velocity \( \omega_0 = 8 \, \text{rad/s} \).
- The angular acceleration \( \alpha = 1.6 \, \text{rad/s}^2 \).

Would you like any more details or have any further questions?

Here are 5 related questions for further exploration:
1. What would happen to the final angular velocity if the angular acceleration were doubled?
2. How much time would it take for the wheel to reach a certain angular displacement with a different initial velocity?
3. What is the relationship between angular velocity and angular acceleration?
4. How does the moment of inertia affect angular acceleration in rotational motion?
5. How can we calculate the total angle described in a similar situation but with changing angular acceleration?

**Tip:** Remember, the area under an angular velocity-time graph represents the angular displacement.

A wheel rotates for 5 seconds with a constant angular acceleration and describes during this time 100 radians. It then rotates with a constant angular velocity and during the next five seconds describes 80 radians. Find the initial angular velocity and the angular acceleration.

Learn how to calculate the initial angular velocity and angular acceleration of a wheel rotating with constant angular acceleration, given it describes 100 radians in 5 seconds, and then moves with constant angular velocity to describe 80 radians in the next 5 seconds. Suitable for Grade 11-12 and college-level physics students.

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