Math Problem Statement

Tom just hopped on the edge of a merry-go-round. What are his linear and angular speeds if the diameter of the merry-go-round is 99 feet and it takes 55 seconds for it to make a complete revolution? Round the solutions to two decimal places.

Solution

To determine Tom's linear speed (vv) and angular speed (ω\omega), we can use the following formulas:

  1. Linear speed:
    v=rωv = r \cdot \omega where rr is the radius of the merry-go-round and ω\omega is the angular speed in radians per second.

  2. Angular speed:
    ω=θt\omega = \frac{\theta}{t} where θ=2π\theta = 2\pi radians (a full revolution) and tt is the time for one revolution.

Step 1: Find the radius of the merry-go-round

The diameter is given as 9999 feet. The radius is: r=diameter2=992=49.5feet.r = \frac{\text{diameter}}{2} = \frac{99}{2} = 49.5 \, \text{feet}.

Step 2: Calculate the angular speed

The angular speed ω\omega is: ω=2πt=2π55.\omega = \frac{2\pi}{t} = \frac{2\pi}{55}. ω0.1142radians per second.\omega \approx 0.1142 \, \text{radians per second}.

Step 3: Calculate the linear speed

The linear speed vv is: v=rω=49.50.1142.v = r \cdot \omega = 49.5 \cdot 0.1142. v5.65feet per second.v \approx 5.65 \, \text{feet per second}.

Final Results:

  1. Angular speed: 0.11radians per second0.11 \, \text{radians per second} (rounded to 2 decimal places).
  2. Linear speed: 5.65feet per second5.65 \, \text{feet per second} (rounded to 2 decimal places).

Would you like more details or explanations about these calculations? Here are 5 related questions for further exploration:

  1. What happens to Tom's linear speed if the diameter of the merry-go-round doubles?
  2. How would the angular speed change if the revolution time decreased to 45 seconds?
  3. Can you calculate the total distance Tom travels in one revolution?
  4. What is the relationship between linear speed and the radius of the circle?
  5. If Tom increases his distance from the center by hopping outward, how does his linear speed change?

Tip: Always double-check units when dealing with angular and linear speeds, especially if switching between radians, degrees, or revolutions.

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Math Problem Analysis

Mathematical Concepts

Linear Speed
Angular Speed
Circular Motion

Formulas

Linear Speed: v = r * ω
Angular Speed: ω = θ / t

Theorems

Relationship between linear and angular speed
Formula for circular motion

Suitable Grade Level

Grades 9-12