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Centripetal acceleration is the rate of change of tangential velocity for an object moving along a circular path. This acceleration is always directed towards the center of the circle. Imagine swinging a ball on a string in a circle around your head. The ball is constantly being pulled towards you by the tension in the string.

In our exercise, the formula to calculate centripetal acceleration is given by:

• \[ a = \frac{v^2}{r} \]
• where \( v \) is the speed of the tip, and \( r \) is the radius of the circle.

Based on the calculation, the speed of the fan blade's tip is about 17.2788 m/s, and the radius is 0.15 m. Substituting these values, the acceleration works out to approximately 1992.54 m/s².

This high centripetal acceleration is why the tip of the fan must be designed to withstand strong forces as it spins. All circular motion requires a centripetal force, which in this case, is provided by the rigidity of the fan blade.

To find out how far the tip of a fan blade travels in one complete revolution, we calculate the circumference. The circumference of a circle is the distance around it. For circular objects, this is given by the formula:

• \[ C = 2 \pi r \]
• where \( r \) is the radius.

For the fan with a blade radius of 0.15 m, the calculation becomes:

• \[ C = 2 \pi \times 0.15 = 0.3 \pi \text{ meters} \]

Understanding circumference is crucial for knowing how far the tip travels in one revolution. This value also helps us calculate how many times the fan can perform certain tasks and its efficiency. Knowing the total distance traveled over time can translate into understanding the mechanical wear the fan undergoes.

The period of motion (\( T \)) is the time taken for one complete revolution or cycle of the fan blade. It is essentially the inverse of the frequency, which tells us how many cycles occur in a given time period.

• The formula for calculating the period is: \[ T = \frac{1}{\text{frequency}} \]

In this exercise, the fan completes 1100 revolutions every minute. First, we convert that frequency to revolutions per second, giving us:

• \( \frac{1100}{60} \text{ revolutions per second.} \)

The period is thus:

• \[ T = \frac{60}{1100} \approx 0.0545 \text{ seconds} \]

Understanding the period of motion helps in analyzing and predicting the rotational performance of machines and in designing them for particular applications. It's crucial for ensuring that machines operate within safe limits.

Angular velocity (\( \omega \)) describes how fast an object rotates or revolves relative to another point, i.e., how quickly the angular position changes with time. The fan's angular velocity lets us know the rotational speed in terms of angles rather than linear velocity.

• Angular velocity (\( \omega \)) is expressed in \( \text{radians per second} \).
• It is calculated using the relation between linear speed and radius: \[ v = r \times \omega \]
• Solving for angular velocity gives: \[ \omega = \frac{v}{r} \]

For our fan with a linear velocity of approximately 17.2788 m/s and a radius of 0.15 m, calculate angular velocity as follows:

• \[ \omega = \frac{17.2788}{0.15} \approx 115.192 \text{ rad/s} \]

Angular velocity helps measure how efficiently a rotating device operates. Engineers use this to design and optimize machinery for a wide range of applications, ensuring it meets desired performance criteria.