# Solve Curvilinear Integral Homework: Raise 200N Bucket 30m w/ 750N Man

• MarcMTL
In summary: The Attempt at a SolutionAlthought this can be solved with conservation of energy (750N+200N)*30m = 28,500J, No it can't because the force varies with t.<snip>Thanks for the help!

## Homework Statement

A 750N man raises a 200N bucket of water 30m up a helicoidal staircase. If we know that the staircase has a 10 meter radius and does exactly 4 complete turns in the 30m span, what was the total work done by the man against gravity?

Also, in the same scenario, what is the work required if the bucket had a small hole so that 50N leaked out at a constant rate while it was raised up the stairs?

## Homework Equations

$$\oint F dr$$

## The Attempt at a Solution

Althought this can be solved with conservation of energy (750N+200N)*30m = 28,500J, I'd like to solve it using a line integral. I've solved the first part, but what's giving me a hard time is the 50N lost at a constant rate. Without the line integral, we can simply figure it out as $$\int \frac{50N}{30M}t$$ with t from 0 to 30, which represents the work with the variable force.

However, how would this be done with the line integral?

For the first part, I found the equation of the staircase:

x=10cos(t) dx=-10sin(t)
y=10sin(t) dy=10cos(t)
z=(30t)/8*Pi dz= 15/(4pi)
(As we want z=30 @ 8pi (4 complete turns))

I obtain: $$\oint 950dx + 950dy +950dz$$, t from 0 to 8pi

I can't seem to figure out how to with the variable mass.

Thanks for the help!

Marc

first $\oint F dr$ represents an integral over a closed path,

that isn't the case here, otherwise the wok would be zero, so you should use $\int F dr$

that said, shouldn't the integrand be a vector dot product? $\int \vec{F} \bullet \vec{dr}$

This means the translation contirbuting to the work is only the component of the translation in the direction of the force.

You should check your first intergal gives the correct value as compared to conservation of energy & should probably be able to be reduced to the form you quote

Last edited:
MarcMTL said:

## Homework Statement

A 750N man raises a 200N bucket of water 30m up a helicoidal staircase. If we know that the staircase has a 10 meter radius and does exactly 4 complete turns in the 30m span, what was the total work done by the man against gravity?

Also, in the same scenario, what is the work required if the bucket had a small hole so that 50N leaked out at a constant rate while it was raised up the stairs?

## Homework Equations

$$\oint F dr$$

## The Attempt at a Solution

Althought this can be solved with conservation of energy (750N+200N)*30m = 28,500J,

No it can't because the force varies with t.

<snip>

However, how would this be done with the line integral?

For the first part, I found the equation of the staircase:

x=10cos(t) dx=-10sin(t)
y=10sin(t) dy=10cos(t)
z=(30t)/8*Pi dz= 15/(4pi)
(As we want z=30 @ 8pi (4 complete turns))

So you have

$$\vec R(t) = \langle 10\cos(t),10\sin(t),\frac {15}{4\pi}t\rangle$$

which looks correct. Now you just need to write the force vector:

$$\vec F(t) = \langle ?,?,?\rangle$$

Here are some hints:

1. What direction is the force vector in?
2. If the water is 200N when t = 0 and 150N when $t = 8\pi$, and it loses weight at a constant rate, can you figure out an equation for W(t)?

## 1. How do I solve a curvilinear integral?

To solve a curvilinear integral, you need to first determine the limits of integration, which are the starting and ending points of the curve. Then, you need to find the equation of the curve and integrate it with respect to the variable you are given. This will give you the solution to the integral.

## 2. What is the purpose of raising a 200N bucket 30m with a 750N man?

The purpose of raising a 200N bucket 30m with a 750N man is to perform work against the force of gravity. By raising the bucket, the man is doing work and transferring energy to the system.

## 3. How do the forces involved affect the solution to the integral?

The forces involved, such as the weight of the bucket and the man, will affect the solution to the integral by changing the limits of integration and the equation of the curve. These forces also determine the amount of work being done, which is represented by the integral.

## 4. Are there any simplifications that can be made to the problem?

Yes, there are a few simplifications that can be made to the problem. One is assuming that the weight of the bucket and the man are constant throughout the 30m distance. Another is assuming that the force of gravity is constant. These simplifications can make the problem easier to solve, but may not accurately represent real-world situations.

## 5. What are some real-life applications of solving curvilinear integrals?

Solving curvilinear integrals can be applied to various real-life situations, such as calculating the work done by a crane lifting heavy objects, determining the displacement of a car traveling along a curved path, or finding the energy required to move an object against a force. These calculations are important in engineering, physics, and other scientific fields.