# Solving the Hardest Work Problem Homework

• derek181
In summary, the problem involves a bucket of water with a mass of 100kg attached to a cable with a mass per unit length of 5kg/m. The other end of the cable is attached to a windlass 100m above the bucket. As the bucket is raised at a constant speed, water runs out through a hole in the bottom at a constant rate, causing the mass of the bucket to decrease to 80kg when it reaches the top. A pigeon of mass 2kg lands on the bucket when it is 50m above the ground and starts splashing water at a rate of 1kg/m. To find the work done by the windlass in raising the bucket 100m, you need to evaluate
derek181

## Homework Statement

A bucket of water with mass 100kg is on the ground attached to one end of a cable with mass per unit length of 5kg/m. The other end of the cable is attached to a windlass 100m above the bucket. if the bucket is raised at a constant speed, water runs out through a hole in the bottom at a constant rate to the extent that the bucket would have mass 80kg when it reaches the top. To further complicate matters, a pigeon of mass 2kg lands on the bucket when it is 50m above the ground. He immediately begins taking a bath, splashing water over the side of the bucket at the rate of 1kg/m. Find the work done by the windlass in raising the bucket 100m.

W=FD
∫Fydy

## The Attempt at a Solution

No idea, Can someone help me solve this?

Ah here is an attempt.

1000(0.2kg/m)(9.81m/s2)(100-y)dy+∫1000(100kg)(9.81m/s2)(100-y)dy+∫10050(2kg)(9.81m/s2)(100-y)dy+∫10050(1kg/m)(9.81m/s2)(100-y)dy

derek181 said:

## Homework Statement

A bucket of water with mass 100kg is on the ground attached to one end of a cable with mass per unit length of 5kg/m. The other end of the cable is attached to a windlass 100m above the bucket. if the bucket is raised at a constant speed, water runs out through a hole in the bottom at a constant rate to the extent that the bucket would have mass 80kg when it reaches the top. To further complicate matters, a pigeon of mass 2kg lands on the bucket when it is 50m above the ground. He immediately begins taking a bath, splashing water over the side of the bucket at the rate of 1kg/m. Find the work done by the windlass in raising the bucket 100m.

W=FD
∫Fydy

## The Attempt at a Solution

No idea, Can someone help me solve this?

You are dealing with a variable-mass problem. See, eg., http://en.wikipedia.org/wiki/Variable-mass_system or
http://physics.stackexchange.com/questions/53980/second-law-of-Newton-for-variable-mass-systems
or http://vixra.org/pdf/1309.0210v1.pdf . These can be quite tricky.

I looked at those links and I think you may be overcomplicating the problem. For the parts that are decreasing I found out that I have to take the initial weight minus the rate of decrease multiplied by the length (y). So (100kg-0.2kg/m(ym))dy. I get the correct order of magnitude but the answer is slightly off.

This problem is a type where you can solve by evaluating the definite integral.

derek181 said:
Ah here is an attempt.

1000(0.2kg/m)(9.81m/s2)(100-y)dy+∫1000(100kg)(9.81m/s2)(100-y)dy+∫10050(2kg)(9.81m/s2)(100-y)dy+∫10050(1kg/m)(9.81m/s2)(100-y)dy
I don't seem to be able to match those integrals up with the various components, perhaps because some are erroneous.
Please separate out and identify each of the work components and state the integral for each.

## What is the hardest work problem?

The hardest work problem is a mathematical problem that involves calculating the amount of work done by a force over a certain distance or time. It is considered difficult because it requires a combination of physics and calculus concepts to solve.

## Why is it important to solve the hardest work problem?

Solving the hardest work problem is important because it helps us understand the relationship between force, distance, and work. It also allows us to accurately calculate the amount of work done in various real-life situations, such as lifting objects or pushing a car.

## What are some tips for solving the hardest work problem?

Some tips for solving the hardest work problem include identifying the given information, drawing a diagram or graph to visualize the problem, and using the appropriate formula or equation. It is also helpful to break the problem down into smaller, more manageable steps.

## What are common mistakes when solving the hardest work problem?

Common mistakes when solving the hardest work problem include using the wrong formula or equation, not considering all of the given information, and making calculation errors. It is important to double-check your work and make sure you are using the correct units for each variable.

## How can understanding the hardest work problem benefit us in other areas?

Understanding the hardest work problem can benefit us in other areas by improving our problem-solving skills, developing our understanding of mathematical concepts, and helping us apply these skills to other real-life situations. It can also prepare us for more advanced topics in physics and calculus.

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