Find the linear speed v for each of the following.

the tip of a propeller 3 m long, rotating 500 times per min (Hint: r = 1.5 m)

Verified step by step guidance

Identify the radius of the circular path. Since the propeller is 3 m long, the radius \( r \) is half of that, so \( r = 1.5 \) meters, as given.

Convert the rotational speed from revolutions per minute (rpm) to angular velocity in radians per second. Use the formula \( \omega = 2\pi \times \text{(rotations per second)} \). Since the propeller rotates 500 times per minute, first convert to rotations per second by dividing by 60.

Calculate the linear speed \( v \) using the relationship between linear speed and angular velocity: \( v = r \times \omega \), where \( r \) is the radius and \( \omega \) is the angular velocity in radians per second.

Substitute the values of \( r \) and \( \omega \) into the formula to express \( v \) in terms of known quantities.

Simplify the expression to find the linear speed \( v \) at the tip of the propeller, which represents how fast the tip moves along its circular path.

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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 measures how fast an object rotates, typically in radians per second. It is related to the number of rotations per minute (rpm) by converting revolutions to radians (1 revolution = 2π radians) and minutes to seconds. This conversion is essential to link rotational speed to linear speed.

Radius in Circular Motion

The radius is the distance from the center of rotation to the point of interest on the rotating object. In this problem, the radius is half the length of the propeller (1.5 m). The radius determines the path length traveled by the tip and is crucial for calculating linear speed.

Linear Speed in Circular Motion

Linear speed (v) is the distance traveled per unit time along the circular path and is related to angular velocity (ω) by the formula v = ωr. This means the linear speed increases with both the angular velocity and the radius, representing how fast the tip of the propeller moves through space.

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