Math Problem Statement

Find the linear speed v for the tip of a propeller 3 m long, rotating 400 times per minute (Hint: r = 1.5 m).

Solution

The problem asks us to find the linear speed vv for the tip of a propeller that is 3 meters long, rotating at 400 revolutions per minute. The hint suggests using r=1.5mr = 1.5 \, \text{m} as the radius.

Step-by-step Solution:

  1. Determine the Circumference: The propeller rotates in a circle. The formula for the circumference of a circle is: C=2πrC = 2\pi r where r=1.5mr = 1.5 \, \text{m}.

    Substituting: C=2π×1.5=3πmC = 2\pi \times 1.5 = 3\pi \, \text{m}

  2. Calculate the Linear Speed: Linear speed vv can be determined using the formula: v=circumference×number of rotations per minutev = \text{circumference} \times \text{number of rotations per minute} Given the number of rotations per minute is 400: v=3π×400=1200πm/minv = 3\pi \times 400 = 1200\pi \, \text{m/min}

Thus, the linear speed is: v=1200πm/minv = 1200\pi \, \text{m/min}

Would you like any more details or have questions?

Related Questions:

  1. How does changing the radius affect the linear speed?
  2. If the number of rotations per minute is halved, what happens to the linear speed?
  3. What would be the linear speed in meters per second?
  4. Can you calculate the angular speed of the propeller?
  5. How would the result change if the propeller length increased to 4 meters?

Tip: The linear speed at the tip of a rotating object is directly proportional to both its radius and the number of rotations per unit time.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Circular Motion
Linear Speed
Revolutions per Minute (RPM)

Formulas

v = 2πr × RPM

Theorems

-

Suitable Grade Level

High School (Grades 10-12)