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 () and angular speed (), we can use the following formulas:
-
Linear speed:
where is the radius of the merry-go-round and is the angular speed in radians per second. -
Angular speed:
where radians (a full revolution) and is the time for one revolution.
Step 1: Find the radius of the merry-go-round
The diameter is given as feet. The radius is:
Step 2: Calculate the angular speed
The angular speed is:
Step 3: Calculate the linear speed
The linear speed is:
Final Results:
- Angular speed: (rounded to 2 decimal places).
- Linear speed: (rounded to 2 decimal places).
Would you like more details or explanations about these calculations? Here are 5 related questions for further exploration:
- What happens to Tom's linear speed if the diameter of the merry-go-round doubles?
- How would the angular speed change if the revolution time decreased to 45 seconds?
- Can you calculate the total distance Tom travels in one revolution?
- What is the relationship between linear speed and the radius of the circle?
- 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
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