Math Problem Statement
A 0.60-kg rock is swung in a circular path and in a vertical plane on a 0.25-m length string. At the top of the path, the angular speed is 12.0 rad/s. What is the tension in the string at that point?
Solution
To find the tension in the string at the top of the circular path, we can use the following principles of circular motion:
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Centripetal Force: This is the force required to keep the rock moving in a circular path. At the top of the path, the centripetal force is provided by the tension in the string and the weight of the rock.
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Formula for Centripetal Force: where:
- is the mass of the rock,
- is the linear speed of the rock, and
- is the radius of the circular path.
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Relating Angular Speed and Linear Speed: The linear speed is related to the angular speed by the formula:
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Tension Calculation at the Top of the Path: At the top, the tension in the string and the gravitational force both act downward, providing the centripetal force. Thus, we have:
Now, let’s calculate each part:
Step 1: Given Values
- Mass of rock,
- Radius of circular path,
- Angular speed,
- Gravitational acceleration,
Step 2: Find the Linear Speed
Using the formula :
Step 3: Calculate Centripetal Force
Now, substitute into the centripetal force formula:
Step 4: Solve for Tension
Using , rearrange to solve for :
Final Answer
The tension in the string at the top of the path is approximately:
Would you like further details or have any questions?
Here are 5 related questions to deepen understanding:
- How would the tension change if the angular speed were increased?
- What happens to the tension if the string length is doubled?
- How does the tension vary as the rock moves through different points in the vertical circle?
- What role does gravitational force play in this scenario?
- How would the calculations change if the rock were swung in a horizontal plane instead?
Tip: When solving circular motion problems, always account for the direction of forces, especially gravity, which may either add to or reduce the required tension depending on the point in the motion.
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Math Problem Analysis
Mathematical Concepts
Circular Motion
Centripetal Force
Gravitational Force
Angular Speed
Formulas
Centripetal Force: F_c = m * v^2 / r
Linear Speed: v = ω * r
Tension at the top: T + mg = F_c
Theorems
Newton's Second Law of Motion
Suitable Grade Level
Grades 11-12
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