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The problem in Russian reads:

**161.** Вентилятор вращается с постоянной скоростью и за две минуты совершает 2400 оборотов. Определите частоту вращения вентилятора, период обращения и линейную скорость точки, расположенной на краю лопасти вентилятора на расстоянии 10 см от оси вращения.

Translation:
A fan rotates at a constant speed, making **2400 revolutions in 2 minutes**. Determine the **rotational frequency**, **period of rotation**, and the **linear speed** of a point located at the edge of a blade, 10 cm from the axis of rotation.

---

### **Solution:**

#### 1. Rotational Frequency (Frequency \(f\))
The frequency \(f\) is the number of revolutions per second.

Given:
- **2400 revolutions** in **2 minutes**.

First, convert 2 minutes to **seconds**:
\[
2 \, \text{minutes} = 2 \times 60 = 120 \, \text{seconds}.
\]
Now, calculate the frequency:
\[
f = \frac{2400 \, \text{revolutions}}{120 \, \text{seconds}} = 20 \, \text{revolutions per second} \, (\text{Hz}).
\]

#### 2. Period of Rotation (\(T\))
The period \(T\) is the time for one complete revolution:
\[
T = \frac{1}{f} = \frac{1}{20} = 0.05 \, \text{seconds}.
\]

#### 3. Linear Speed (\(v\))
The linear speed \(v\) of a point on the blade is related to the **radius** and **angular velocity**.

The radius \(r\) is given as:
\[
r = 10 \, \text{cm} = 0.1 \, \text{m}.
\]

Next, we need the **angular velocity** \( \omega \) in radians per second:
\[
\omega = 2\pi f = 2\pi \times 20 = 40\pi \, \text{rad/s}.
\]

Now, use the formula for linear speed:
\[
v = r \cdot \omega = 0.1 \cdot 40\pi = 4\pi \, \text{m/s} \approx 12.57 \, \text{m/s}.
\]

---

### **Final Answers:**
1. **Rotational frequency**: \(20 \, \text{Hz}\).
2. **Period of rotation**: \(0.05 \, \text{seconds}\).
3. **Linear speed**: \(4\pi \, \text{m/s} \approx 12.57 \, \text{m/s}\).

---

Do you need any further clarification or additional help?

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**Related Questions:**
1. What is the difference between angular velocity and linear velocity?
2. How is frequency related to angular speed?
3. What are some practical applications of calculating linear speed in rotating systems?
4. How would the linear speed change if the radius of rotation were doubled?
5. How would the answers change if the fan performed 2400 rotations in 1 minute?

**Tip:** When dealing with rotational motion, always double-check units to ensure that time and angles are consistent (e.g., seconds and radians).

A fan rotates at a constant speed, making 2400 revolutions in 2 minutes. Determine the rotational frequency, period of rotation, and the linear speed of a point located at the edge of a blade, 10 cm from the axis of rotation.

Learn how to determine the rotational frequency, period of rotation, and linear speed of a point on a fan blade located 10 cm from the axis, given that the fan makes 2400 revolutions in 2 minutes.

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