Problem 8 - Thomas Calculus in SI Units

Question

A bag of sand originally weighing \(600 \mathrm{N}\) was lifted at a constant rate. As it rose, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to \(6 \mathrm{m} .\) How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Step-by-Step Solution

Answer

2700 Joules of work was done.

Step 1: Understand the Problem

We need to find out the work done to lift a bag of sand to 6 meters, considering that the sand leaks out at a constant rate so that half of it is gone by the time it reaches that height. The work done is the integral of force across the distance.

Step 2: Determine Initial Conditions

Initially, the weight of the sand in the bag is 600 N. By the time it reaches 6 m, it weighs 300 N because the sand is leaking linearly, reducing the weight to half of its initial value.

Step 3: Define the Force Function

Since the sand leaks at a constant rate while the bag is lifted, the weight of the sand as a function of height, h, is a linear function. Initially, at h = 0, the force is 600 N, and at h = 6 m, the force is 300 N. We can express this linearly as: \[ F(h) = 600 - 50h \]

Step 4: Calculate the Work Done

The work done to lift the sand can be expressed as the integral of the force over the distance, from 0 to 6 m:\[ W = \int_{0}^{6} F(h) \, dh = \int_{0}^{6} (600 - 50h) \, dh \]

Step 5: Integrate the Force Function

Calculate the integral:\[ W = \int_{0}^{6} (600 - 50h) \, dh = \left[ 600h - 25h^2 \right]_{0}^{6} \] Substitute the boundaries:\[ W = (600 \times 6 - 25 \times 6^2) - (600 \times 0 - 25 \times 0^2) \] \[ W = 3600 - 25 \times 36 \] \[ W = 3600 - 900 \] \[ W = 2700 \text{ J} \]

Step 6: Conclusion

The total work done in lifting the bag of sand to 6 meters, considering the constant leak, is 2700 Joules.

Key Concepts

Integration in Calculus

Integration in calculus is a core concept used to solve problems involving accumulated quantities, such as work done when the force applied changes over a distance. In this physics problem, we use integration to calculate the total work done while lifting a bag of sand where the weight—and thus, the force—changes as sand leaks.

Integration helps to "sum up" the effects of these continuously changing quantities. For instance, when we lift the sand, the total work needed isn't simply force times distance because the force itself is not constant. **Instead**, we express force as a function of distance and integrate over that distance.

For our exercise, the function representing force is linear, decreasing as the sand leaks out. Calculus allows us to find the precise amount of work by integrating this function over the specified distance. The integration shows how calculus transforms a seemingly complex problem into manageable calculations.

Force as a Function of Distance

In physics, force is often not a constant and can change depending on many factors, including distance. In the problem with the leaking sand, the force exerted at any point directly depends on how much sand remains in the bag as it is lifted.

To express force as a function of height, we begin with the maximum initial force, then reduce this force linearly as height increases and sand continues to leak. In mathematical terms, this can be represented with the function:

• At a height of 0 meters, the force is 600 N.
• At a height of 6 meters, the force drops to 300 N.
• The linear function of force: \( F(h) = 600 - 50h \).

This formula exemplifies how forces in real-world physics problems can depend dynamically on other variables like distance, making them important to model correctly for accurate calculations.

Physics Problem-Solving

Physics problem-solving often involves breaking down the problem into comprehensible parts and applying appropriate mathematical concepts. In the case of this exercise, understanding the physical context is essential: **a bag leaking sand as it's lifted**.

By using problem-solving strategies:

• Determine what you need to find. Here, it's the total work done.
• Express the given data mathematically; we used a function for the force based on height.
• Apply calculus for dynamic problems, integrating to accumulate effects such as the variable force over a distance.

Each step provides more insight, guiding you efficiently from concept to solution. Physics problem-solving leverages conceptual understanding and practical application of formulas, particularly powerful in scenarios involving changing variables, to arrive at precise, meaningful solutions.