Problem 8 - Thomas Calculus

Question

Leaky sandbag A bag of sand originally weighing 144 lb 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 18 \(\mathrm{ft}\) . How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Step-by-Step Solution

Answer

The work done lifting the sand 18 feet is 1944 ft-lb.

Step 1: Understanding the Problem

We're given a bag of sand initially weighing 144 lbs, lifted to 18 feet such that half of the sand leaks out during the lift. We need to calculate the work done, neglecting the weight of the bag and equipment.

Step 2: Identifying Variables

Let the initial weight of the sand be \( W_0 = 144 \space \text{lb} \), and the height it is lifted to be \( h = 18 \space \text{ft} \). By the time it reaches this height, the weight of the sand has reduced to half, which is \( W = 72 \space \text{lb} \).

Step 3: Determine the Rate of Loss

Since the sand leaks at a constant rate till it is half empty at 18 ft, we use the fact that weight decreases linearly from \( 144 \space \text{lb} \) to \( 72 \space \text{lb} \). The constant rate of sand loss is therefore \( \frac{72}{18} = 4 \space \text{lb/ft} \).

Step 4: Setting Up the Integral for Work

The weight of the sand at any height \( x \) from 0 to 18 ft is \( 144 - 4x \). The force at any height is \( (144 - 4x) \) lb, and the work done is the integral of the force over the distance from 0 to 18 ft.

Step 5: Calculating Work Done

The work done \( W \) is given by the integral:\[W = \int_{0}^{18} (144 - 4x)\, dx\]Evaluate this integral to find the work done.

Step 6: Evaluating the Integral

Let's integrate:\[W = \int_{0}^{18} (144 - 4x)\, dx = \left[144x - 2x^2\right]_{0}^{18}\]Substitute the limits:\[W = \left(144 \times 18 - 2 \times 18^2\right) - \left(144 \times 0 - 2 \times 0^2\right) = 2592 - 648\]Thus, \( W = 1944 \space \text{ft-lb} \).

Key Concepts

Work Done

When we talk about "work done" in physics, we refer to the amount of energy transferred by a force acting over a distance. This principle is important in various fields including mechanics and engineering. Work can be calculated using the formula:\[ W = F \times d \]where:

• \( W \) represents the work done,
• \( F \) is the force applied, and
• \( d \) is the displacement or distance over which the force is applied.

For the exercise with the leaky sandbag, calculating the work done involves considering how the force changes as the sand leaks. Initially, the bag has more weight, and that weight decreases steadily as sand leaks out. Calculating work in this scenario means we need to account for this changing force as the weight of the sandbag reduces with height.
Ultimately, the work done in lifting the sandbag is the integral of the changing force over the height raised, which is resolved through calculus to find the total energy expended in lifting the bag to a certain height.

Integral Calculus

Integral calculus is a key tool for analyzing problems involving continuously changing quantities, like the weight of the sandbag in our exercise. The act of integrating allows us to sum up small quantities, providing a total over a range, which in this case, translates to calculating the work done.In the exercise, we set up an integral to account for the changing force as the sandbag ascends. The integral used is:\[ W = \int_{0}^{18} (144 - 4x)\, dx \]This integral considers:

• A variable force \( (144 - 4x) \), accounting for the loss of sand weight,
• The limits of integration from 0 to 18 feet, which covers the total distance the sandbag is raised.

By solving this integral, we "add up" the small amounts of work done over each infinitesimally small segment of the 18 feet, arriving at the total work done, calculated as 1944 ft-lb. This method highlights how calculus provides a powerful framework to tackle such changing processes.

Linear Rate of Change

In the context of calculus and the given problem, a "linear rate of change" refers to how a quantity changes at a consistent, unchanging rate over time or space. For the leaky sandbag, the rate at which sand is leaking out is constant. The problem provides a scenario where the sand leaks at a steady pace as the bag is lifted. Knowing that the weight reduced from 144 lb to 72 lb over 18 feet allows us to derive a constant rate:

• Initial weight of sand: 144 lb
• Final weight of sand at 18 feet: 72 lb
• Constant rate of sand loss: 4 lb/ft, as calculated from the linear decline over the height.

This constant loss rate represents a simple linear relationship because the sandbag's weight decreases evenly as it is hoisted upwards. Calculating this rate is crucial because it lets us accurately evaluate how the force applied changes linearly, enabling us to set up the integral appropriately and solve for total work done effectively.