A fan blade of length 1/root pi mtere rotates with frequency 5 cycle /sec perpen

To analyze the motion of a fan blade with a given length and frequency, we can break down the problem into manageable parts. The fan blade's length is \( \frac{1}{\sqrt{\pi}} \) meters, and it rotates at a frequency of 5 cycles per second, which is equivalent to 5 Hz. Let's explore how to calculate the important parameters related to this rotating blade, such as its angular velocity and linear velocity.

Understanding Angular Velocity

Angular velocity is a measure of how quickly an object rotates. It is typically expressed in radians per second. Since the fan blade completes 5 cycles per second, we can convert this frequency into angular velocity using the formula:

• Angular Velocity (ω) = 2π × Frequency (f)

Substituting the given frequency:

• ω = 2π × 5 = 10π radians/second

Calculating Linear Velocity

Next, we can find the linear velocity of the tip of the fan blade. Linear velocity (v) can be calculated using the formula:

• Linear Velocity (v) = Angular Velocity (ω) × Radius (r)

In this case, the radius is the length of the fan blade, which is \( \frac{1}{\sqrt{\pi}} \) meters. Plugging in the values:

• v = 10π × \( \frac{1}{\sqrt{\pi}} \)
• v = \( \frac{10π}{\sqrt{\pi}} \) = 10√π meters/second

Summary of Key Findings

To summarize, we have determined the following:

• The angular velocity of the fan blade is \( 10π \) radians/second.
• The linear velocity at the tip of the fan blade is \( 10\sqrt{\pi} \) meters/second.

These calculations provide a clear understanding of the dynamics of the rotating fan blade. If you have any further questions or need clarification on any of these concepts, feel free to ask!