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

1. Identify the given values: the radius \( r = 1.5 \) m and the number of rotations per minute \( n = 500 \).
2. Recall the formula for the circumference of a circle: \( C = 2\pi r \).
3. Calculate the distance traveled in one rotation using the circumference: \( C = 2\pi \times 1.5 \).
4. Determine the total distance traveled in one minute by multiplying the circumference by the number of rotations: \( D = C \times 500 \).
5. Use the formula for linear speed \( v = \frac{D}{t} \), where \( t = 1 \) minute, to find the linear speed.

Key Concepts

Linear Speed

Linear speed refers to the distance traveled per unit of time by a point on a rotating object. It is calculated using the formula v = rω, where v is the linear speed, r is the radius, and ω is the angular speed in radians per second. Understanding linear speed is crucial for solving problems involving circular motion.

Angular Speed

Angular speed is the rate of rotation of an object, typically measured in radians per second. It can be derived from the number of rotations per minute (RPM) by converting it to radians using the formula ω = (2π * RPM) / 60. This concept is essential for relating the rotational motion of an object to its linear motion.

Radius of Rotation

The radius of rotation is the distance from the center of rotation to the point of interest on the rotating object. In this case, the propeller's length is given as 3 m, but the radius for calculating linear speed is half of that, or 1.5 m. This concept is fundamental in determining how far a point on the object travels during rotation.