Math Problem Statement
Solution
To solve this problem, we can use the principle of work and energy in rotational motion. Here's the detailed step-by-step solution:
Given Data
- Moment of inertia of the top,
- Tension in the string,
- Length of string pulled, (convert to meters)
- Initial angular speed, (since the top is initially at rest)
Solution Approach
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Calculate the Work Done by the Force (Tension) on the Top: Since the force is acting through a distance , the work done by the force is:
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Relate Work Done to Rotational Kinetic Energy: The work done by the force goes into the rotational kinetic energy of the top. The rotational kinetic energy of a rotating object is given by: where is the angular speed we want to find.
According to the work-energy principle: Substituting and :
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Solve for : Rearrange the equation to solve for :
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Substitute the Values: Calculate each step:
Answer
The angular speed of the top after 114 cm of string has been pulled is approximately:
Would you like more details on any part of the solution?
Additional Related Questions:
- How would the solution change if the tension in the string varied with time?
- What is the relationship between linear and angular acceleration in this setup?
- How would we calculate the angular speed if the top had an initial angular velocity?
- What role does moment of inertia play in determining the angular speed?
- How would the outcome change if there were frictional forces acting on the top?
Tip:
In rotational dynamics, always consider the work-energy principle as a powerful tool to relate forces, displacements, and rotational speeds.
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Math Problem Analysis
Mathematical Concepts
Rotational Dynamics
Work-Energy Principle
Moment of Inertia
Formulas
Work Done: W = F × s
Rotational Kinetic Energy: KE_rot = 1/2 I ω^2
Angular Speed: ω = √(2Fs / I)
Theorems
Work-Energy Theorem in Rotational Motion
Suitable Grade Level
Undergraduate Physics
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