Math Problem Statement
solve
Solution
To solve this problem, we calculate the work done lifting the bucket with the mass decreasing as the sand leaks out. Here's the detailed solution:
Given:
- Initial mass of bucket and sand =
- Height =
- Rate of sand leakage =
- Gravitational acceleration =
Step-by-step process:
1. Mass as a function of height
The bucket loses for every meter lifted. Hence, the mass at a height is:
2. Work done lifting the bucket
The work done in lifting an object is given by:
Substitute :
3. Solve the integral
Expand and integrate:
Evaluate each term:
Substitute back:
Final Answer:
The work done lifting the bucket is 3062.5 Joules.
Would you like more clarification on any step? Here are related questions you might find helpful:
- How does sand leakage affect the total work done?
- What is the physical significance of integrating mass as a function of height?
- How would the answer change if the sand leaked at a different rate?
- How can this problem be visualized with a graph of mass vs height?
- Can we derive an alternative method without calculus?
Tip: Always consider variable mass problems using integrals to account for gradual changes systematically.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Work and Energy
Integration
Formulas
Work = ∫ Force × Distance
Force = Mass × Gravitational Acceleration
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grade 11-12
Related Recommendation
Calculate Force and Work for a Leaking Bucket Problem
Work and Force Calculation for a Leaking Bucket Lift Problem
Work Done to Lift a Leaking Bucket Using Variable Mass and Force Functions
Work Calculation for Lifting a Bucket with Variable Mass and Rope
Work Integral Setup for Lifting a Bucket from a Well with Varying Weight