Trouble with this integral and only an hour to solve it

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In summary, the question is about calculating the tension in a cable attached to a deep-sea diver in Loch Ness. Using the formula F(x) = (\mu*x+m)*g-\rho_{water}*g*((d/2)^2*pi*x+V), the tension can be calculated a distance above the diver. The next step involves finding the time it takes for a signal to reach the surface, which can be done by integrating the differential equation \frac{dx}{dt}=\sqrt{ax+b}. The starting condition is x=0 at t=0 and the length of the cable is h. The final solution is T, which is the time needed for the signal to reach the surface.
  • #1
Brianjw
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Well first here is the question:

A deep-sea diver is suspended beneath the surface of Loch Ness by a cable of length h that is attached to a boat on the surface . The diver and his suit have a total mass of m and a volume of V. The cable has a diameter of d and a linear mass density of mu. The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat.

Calculate the tension in the cable a distance above the diver. The buoyant force on the cable must be included in your calculation. Take the free fall acceleration to be g.

I solve this one, I got:

[tex] F(x) = (\mu*x+m)*g-\rho_{water}*g*((d/2)^2*pi*x+V) [/tex]

For the next one I can't solve it, the integral is nuts for me, I must be overlooking something

The speed of transverse waves on the cable is given by v = sqrt(F/mu). The speed therefore varies along the cable, since the tension is not constant. (This expression neglects the damping force that the water exerts on the moving cable.) Integrate to find the time required for the first signal to reach the surface. Take the free fall acceleration to be g.

From what I understand I have to plug in my F(x) from part b and integrate, but I'm not sure how to integrate it since its just so huge. Any ideas?

Thanks!

Brian
 
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  • #2
Brianjw said:
Well first here is the question:



[tex] F(x) = (\mu*x+m)*g-\rho_{water}*g*((d/2)^2*pi*x+V) [/tex]

For the next one I can't solve it, the integral is nuts for me, I must be overlooking something

The speed of transverse waves on the cable is given by v = sqrt(F/mu). The speed therefore varies along the cable, since the tension is not constant. (This expression neglects the damping force that the water exerts on the moving cable.) Integrate to find the time required for the first signal to reach the surface. Take the free fall acceleration to be g.

From what I understand I have to plug in my F(x) from part b and integrate, but I'm not sure how to integrate it since its just so huge. Any ideas?

Thanks!

Brian
Write [tex]F/\mu[/tex] in the form [tex]F/\mu=ax+b[/tex] .

Remember that

[tex]v=x'=\frac{dx}{dt}[/tex] .

So you have got the differential equation

[tex]\frac{dx}{dt}=\sqrt{ax+b}[/tex]

with the starting condition x=0 at t=0.
You can integrate it by separating the variables:

[tex]\int_0^h\frac{dx}{\sqrt{ax+b}}=\int_0^T{dt}=T[/tex]

where T is the time needed for the signal to reach the surface and h is the length of the cable. The integrand at the left side is

[tex](ax+b)^{-1/2}[/tex] .

I think you can do it...

ehild
 
  • #3


Hi Brian,

I understand that you are struggling with the second part of this problem and only have an hour to solve it. I can imagine the pressure you must be feeling, but don't worry, I'm here to help.

First of all, great job on solving the first part! It looks like you have a good understanding of the problem and have found the correct expression for the tension in the cable.

For the second part, you are correct that you need to integrate your expression for F(x) over the length of the cable to find the time it takes for the signal to reach the surface. I can see why this integral may seem daunting, but don't worry, there are ways to make it more manageable.

One approach you can take is to break up the integral into smaller, more manageable pieces. For example, you can split it into two integrals, one for the first term and one for the second term in your expression for F(x). You can also try using substitution or integration by parts to simplify the integral.

Another approach is to use numerical integration, where you use a computer program or calculator to approximate the integral. This can be a quicker and more efficient method, especially if you are short on time.

I recommend trying these different approaches and see which one works best for you. Don't forget to also include the buoyant force in your calculation. And remember, don't get too caught up in the details and take a step back if you feel stuck. Sometimes a fresh perspective can help.

Good luck and I hope you are able to solve the integral in time!
 

Related to Trouble with this integral and only an hour to solve it

1. How do I approach solving a difficult integral in a limited amount of time?

When faced with a challenging integral that needs to be solved in a short period of time, it is important to first understand the properties and rules of integration. This can help you identify any patterns or techniques that can be used to simplify the integral. It may also be helpful to break the integral into smaller parts and solve them individually before combining them back together.

2. What are some common techniques for solving integrals quickly?

Some common techniques for solving integrals quickly include using substitution, integration by parts, and trigonometric identities. It is also important to simplify the integral as much as possible before attempting to solve it. Additionally, having a good understanding of algebra and mathematical properties can also help in solving integrals efficiently.

3. How do I know if I am on the right track when solving an integral?

One way to check if you are on the right track when solving an integral is to use a calculator or online tool to verify your answer. Another method is to differentiate your solution and see if it matches the original function. If your answer does not match or if you encounter any inconsistencies, it may be necessary to go back and review your steps or try a different approach.

4. What are some common mistakes to avoid when solving integrals under time constraints?

Some common mistakes to avoid when solving integrals under time constraints include rushing through the problem without fully understanding it, making calculation errors, and forgetting to apply important rules or properties. It is important to stay organized and focused while solving the integral, and to double check your work before submitting your final answer.

5. How can I improve my skills in solving integrals quickly?

The best way to improve your skills in solving integrals quickly is to practice regularly and to familiarize yourself with different types of integrals and their corresponding techniques. You can also seek out additional resources, such as textbooks or online tutorials, to learn new methods for solving integrals. It can also be helpful to work with a tutor or study group to exchange ideas and strategies for solving integrals efficiently.

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