Math Problem Statement

solve

A bucket is filled with sand so that the total mass (sand and bucket) is 15 kg. The bucket is to be lifted to the top of a 25-meter tall building by a rope of negligible weight. However, the bucket has a hole in it, and leaks 0.2 kg of sand each meter it is lifted. Find the work done lifting the bucket to the top of the building.

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:

  1. Initial mass of bucket and sand = 15kg15 \, \text{kg}
  2. Height = 25m25 \, \text{m}
  3. Rate of sand leakage = 0.2kg/m0.2 \, \text{kg/m}
  4. Gravitational acceleration = g=9.8m/s2g = 9.8 \, \text{m/s}^2

Step-by-step process:

1. Mass as a function of height

The bucket loses 0.2kg0.2 \, \text{kg} for every meter lifted. Hence, the mass m(h)m(h) at a height hh is:

m(h)=150.2hm(h) = 15 - 0.2h

2. Work done lifting the bucket

The work done WW in lifting an object is given by:

W=025m(h)gdhW = \int_0^{25} m(h) \cdot g \, dh

Substitute m(h)=150.2hm(h) = 15 - 0.2h:

W=025(150.2h)9.8dhW = \int_0^{25} (15 - 0.2h) \cdot 9.8 \, dh

W=9.8025(150.2h)dhW = 9.8 \int_0^{25} (15 - 0.2h) \, dh

3. Solve the integral

Expand and integrate:

W=9.8[02515dh0250.2hdh]W = 9.8 \left[ \int_0^{25} 15 \, dh - \int_0^{25} 0.2h \, dh \right]

W=9.8[15h0250.2h22025]W = 9.8 \left[ 15 \cdot h \big|_0^{25} - 0.2 \cdot \frac{h^2}{2} \big|_0^{25} \right]

Evaluate each term:

  • 02515dh=1525150=375\int_0^{25} 15 \, dh = 15 \cdot 25 - 15 \cdot 0 = 375
  • 0250.2hdh=0.225220.2022=0.2312.5=62.5\int_0^{25} 0.2h \, dh = 0.2 \cdot \frac{25^2}{2} - 0.2 \cdot \frac{0^2}{2} = 0.2 \cdot 312.5 = 62.5

Substitute back: W=9.8[37562.5]W = 9.8 \left[ 375 - 62.5 \right]

W=9.8312.5W = 9.8 \cdot 312.5

W=3062.5JW = 3062.5 \, \text{J}


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:

  1. How does sand leakage affect the total work done?
  2. What is the physical significance of integrating mass as a function of height?
  3. How would the answer change if the sand leaked at a different rate?
  4. How can this problem be visualized with a graph of mass vs height?
  5. Can we derive an alternative method without calculus?

Tip: Always consider variable mass problems using integrals to account for gradual changes systematically.

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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