# Finding the right integral

• Ailo
In summary, the problem involves a small mass being pulled to the top of a frictionless halfcylinder by a cord over the top of the cylinder. The first part asks to show that F=mg cos(theta) if the mass moves at a constant speed. The second part involves finding the work done by directly integrating ∫Fds and solving for the work done in moving the mass from the bottom to the top of the half-cylinder. By setting up the integral as ∫mgcos(theta)*Rdθ, the solution can be simplified to mgR.

## Homework Statement

A small mass m is pulled to the top of a frictionless halfcylinder (of radius R) by a cord that passes over the top of the cylinder. (a) If the mass moves at a constant speed, show that $$F=mg cos(\theta)$$. The angle is between the horizontal and the radius drawn to the mass.

(b) By directly integrating
$$\int{Fds}$$
find the work done in moving the mass at constant speed from the bottom to the top of the half-cylinder. Here ds represents an incremental displacement of the small mass.

## The Attempt at a Solution

The a-part was easy when I drew a diagram. The b-part is the one I'm struggling with. With the Work-Energy theorem I get that the work done by F is mgR. But what integral should I compute and why? Have no idea whatsoever..

Ailo said:
(b) By directly integrating
$$\int{Fds}$$
find the work done in moving the mass at constant speed from the bottom to the top of the half-cylinder. Here ds represents an incremental displacement of the small mass.

The a-part was easy when I drew a diagram. The b-part is the one I'm struggling with. With the Work-Energy theorem I get that the work done by F is mgR. But what integral should I compute and why? Have no idea whatsoever..

Hi Ailo!

(I'm not sure what you mean by mgR)

The integral is given to you … ∫F ds, where s is the displacement.

(Remember, the string is always tangent to the cylinder.)

Hi! Thx for the answer, but the problem is how to set it up. I should get an expression, integrate it, and end up with the answer mg*R. I've got the force as a function of the angle, and I don't understand how to integrate it over a distance.

Maybe I didn't explain the situation good enough. The half cylinder lies on the ground, and we pull the mass up along the quartercircle. Does anybody understand? =)

Hi Ailo!

Just decide what ds is (in terms of θ), and then integrate mgcosθ ds, and you should get mgR.

That's the problem. I've never done a problem like this before...

ok …

what is ds in terms of θ?

in other words, if you increase the angle by dθ, how much do you increase the length (s) of the string by?

My best guess is to make a triangle with sides R, R and ds. Will that work?

Ohh! Now I get it. It's (theta)*R, right?

*palmslap

Yup!

ds = Rdθ

## 1. What is an integral and why is it important?

An integral is a mathematical concept that represents the area under a curve on a graph. It is important because it allows us to calculate the total value or quantity of something that is continuously changing.

## 2. How do I know which integral to use for a given problem?

The type of integral to use depends on the function being integrated and the limits of the integration. Common types of integrals include definite and indefinite integrals, as well as single and multiple integrals. It is important to carefully analyze the problem and choose the appropriate integral based on the given information.

## 3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration, meaning it calculates the area under a specific portion of a curve. An indefinite integral, on the other hand, does not have specific limits and instead gives a general result in the form of a function.

## 4. Can I use a calculator to find an integral?

Yes, many calculators have a built-in integral function that can calculate integrals for a given function and limits. However, it is important to understand the process of finding an integral by hand in order to fully grasp the concept.

## 5. Are there any tips or tricks for finding the right integral?

One tip is to try using different integration techniques, such as substitution or integration by parts, to see which one works best for a given problem. It is also important to carefully analyze the problem and make sure all necessary information is included in the integral. Practice and familiarizing oneself with different types of integrals can also help in finding the right one for a given problem.

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