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Let's solve the two problems step by step.

### **Problem 1**
*Un motor gira a \(120 \, \text{rad/s}\) y desacelera a \(60 \, \text{rad/s}\) en 5 segundos. ¿Cuál es su aceleración angular?*

**Solution:**

The angular acceleration (\(\alpha\)) can be calculated using the formula:

\[
\alpha = \frac{\omega_f - \omega_i}{t}
\]

Where:
- \(\omega_i = 120 \, \text{rad/s}\) (initial angular velocity),
- \(\omega_f = 60 \, \text{rad/s}\) (final angular velocity),
- \(t = 5 \, \text{seconds}\) (time).

Plugging in the values:

\[
\alpha = \frac{60 \, \text{rad/s} - 120 \, \text{rad/s}}{5 \, \text{seconds}} = \frac{-60 \, \text{rad/s}}{5 \, \text{seconds}} = -12 \, \text{rad/s}^2
\]

So, the angular acceleration is \(-12 \, \text{rad/s}^2\). The negative sign indicates that the motor is decelerating.

### **Problem 2**
*Una rueda de radio 0.5 tiene una velocidad angular de \(10 \, \text{rad/s}\). Calcula la velocidad tangencial de un punto en el borde de la rueda.*

**Solution:**

The tangential velocity (\(v_t\)) at the edge of a wheel is related to the angular velocity (\(\omega\)) by the equation:

\[
v_t = \omega \times r
\]

Where:
- \(\omega = 10 \, \text{rad/s}\) (angular velocity),
- \(r = 0.5 \, \text{m}\) (radius of the wheel).

Substituting the values:

\[
v_t = 10 \, \text{rad/s} \times 0.5 \, \text{m} = 5 \, \text{m/s}
\]

Thus, the tangential velocity of a point on the edge of the wheel is \(5 \, \text{m/s}\).

---

### Summary of Answers:
1. **Problem 1:** The angular acceleration is \(-12 \, \text{rad/s}^2\).
2. **Problem 2:** The tangential velocity is \(5 \, \text{m/s}\).

Would you like to explore any specific part of these solutions further or have any questions about the concepts used?

Here are some related questions:
1. What would the angular acceleration be if the motor increased its speed from 60 rad/s to 120 rad/s in 5 seconds?
2. How would the tangential velocity change if the radius of the wheel were doubled?
3. What is the relationship between angular acceleration and torque?
4. If the motor continued to decelerate at the same rate, how long would it take to stop completely?
5. How does the tangential velocity relate to centripetal acceleration?
6. What would be the angular velocity if the wheel’s radius was reduced to 0.25 m but maintained the same tangential velocity?
7. How would the problem change if the motor decelerated in 3 seconds instead of 5?
8. Can the tangential velocity of a wheel be different at different points along its radius?

**Tip:** When working with rotational motion, remember that angular quantities (like angular velocity and angular acceleration) are often directly analogous to their linear counterparts. This relationship helps in understanding and solving problems involving rotation.

Explore how to calculate angular acceleration and tangential velocity in rotational motion problems. This detailed explanation covers angular velocity changes and their impact on acceleration, along with practical examples. Suitable for high school students studying physics and mechanics.

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