Math Problem Statement
Solution
To solve this problem, we need to calculate the maximum mass () that the rope can swing without breaking. We are given:
- Maximum tension the rope can handle, .
- Speed of the mass in circular motion, .
- Radius of the circle, .
Since the mass is moving in a circular path, the tension in the rope provides the centripetal force required to keep the mass moving in a circle. The centripetal force needed for circular motion is given by:
Here, the tension in the rope is equal to the centripetal force , so:
Rearranging to solve for :
Substituting the values:
Calculating the denominator first:
Then, the mass :
Answer:
The largest mass the rope can swing at 14.2 m/s in a circle of radius 0.425 m without breaking is approximately 0.64 kg.
Would you like a deeper explanation, or do you have any questions?
Here are some additional related questions you might find helpful:
- What happens to the maximum mass if the speed increases to 20 m/s?
- How does the radius of the circle affect the maximum allowable mass?
- If the rope's breaking tension was increased to 500 N, what would be the new maximum mass?
- How would you calculate the maximum tension for a different rope length?
- How does the concept of centripetal force apply to other circular motion scenarios, like cars on curved roads?
Tip: In circular motion problems, remember that increasing speed significantly increases the required centripetal force because it’s proportional to the square of the velocity ().
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Math Problem Analysis
Mathematical Concepts
Centripetal Force
Tension in Circular Motion
Newton's Second Law
Formulas
Centripetal force formula: F_c = mv^2 / r
Rearranged formula for mass: m = T * r / v^2
Theorems
Newton's Second Law of Motion
Suitable Grade Level
Grades 10-12
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