A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. What is the speed of the tip of a blade 10 s after the fan is turned off?

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Determine the initial angular velocity (ω₀) of the fan in radians per second. Convert the given rotational speed of 60 rpm (revolutions per minute) to radians per second using the formula: ω₀ = (2π × 60) / 60.

Calculate the angular acceleration (α) of the fan as it slows down uniformly to a stop. Use the kinematic equation for rotational motion: ω = ω₀ + αt, where ω is the final angular velocity (0 rad/s at stop), ω₀ is the initial angular velocity, and t is the time to stop (25 s). Solve for α.

Find the angular velocity (ω) of the fan 10 seconds after it is turned off. Use the same kinematic equation: ω = ω₀ + αt, where t = 10 s.

Determine the linear speed (v) of the tip of a blade at this angular velocity. Use the relationship between linear speed and angular velocity: v = rω, where r is the radius of the blade (half the diameter, 80 cm / 2 = 40 cm = 0.4 m).

Substitute the values of r and ω (calculated in the previous steps) into the formula v = rω to find the linear speed of the blade tip 10 seconds after the fan is turned off.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). In this scenario, the ceiling fan's blades initially rotate at 60 rpm, which can be converted to radians per second to analyze the motion of the blades over time.

Linear Speed

Linear speed refers to the distance traveled by a point on a rotating object per unit of time. For a rotating blade, the linear speed at the tip can be calculated using the formula v = rω, where v is the linear speed, r is the radius of the rotation, and ω is the angular velocity. This concept is crucial for determining how fast the tip of the fan blade is moving at any given moment.

Deceleration

Deceleration is the rate at which an object slows down, often expressed in terms of negative acceleration. In this case, after the fan is turned off, it coasts to a stop over 25 seconds, indicating a uniform deceleration. Understanding this concept is essential for calculating the speed of the fan blade at specific times after it has been turned off.

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