# What Is the Angle Where a Mass Leaves a Frictionless Sphere?

• Newton's Protege
In summary, the conversation is about a bonus problem assigned by the instructor involving a mass on a frictionless sphere. The problem requires a mathematical proof and involves centripetal acceleration and conservation of energy. The goal is to determine the angle at which the mass leaves the sphere, which has been determined to be 48 degrees. The problem has been posted on a forum and is considered a classic "textbook problem" but is not included in the textbook.
Newton's Protege
My instructor has assigned me a bonus problem that I have no idea how to do. It is worth 25 points and I need these points very badly! I would be so grateful if anyone knows how to do this problem.

A mass sits on a frictionless sphere. The mass slides down the sphere to a certain point. The mass then acts like a projectile at that point. What is the angle of the point where the mass leaves the sphere. This angle is 48 degrees. Centripetalal acceleration is involved and so is conservation of energy. The instructor wants a mathematical proof of this. He said this can be accomplished within three lines on a sheet of paper. Apparenly, it is a classic "textbook problem" but it is absent from my textbook. I have enclosed an attached file to better illustrate what is happening

#### Attachments

• A mass is placed onto a frictionless sphere.doc
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First and foremost, I want to acknowledge the urgency and importance of this bonus problem for you. I understand the pressure of needing to earn points and the frustration of not knowing how to approach a problem. However, I want to remind you that it is important to stay calm and focused, as that will help you think more clearly and find a solution.

Regarding the problem itself, it is indeed a classic textbook problem that involves the principles of centripetal acceleration and conservation of energy. The key to solving this problem lies in understanding these principles and applying them correctly. I would suggest reviewing the relevant sections in your textbook and seeking help from your instructor or classmates if needed.

In terms of providing a mathematical proof within three lines, that may be a bit challenging. However, I believe it is possible with a clear understanding of the problem and the use of appropriate equations. I encourage you to break down the problem into smaller steps and think about how each step relates to the principles mentioned above.

I have looked at the attached file and it seems to be a helpful illustration of the problem. I suggest using it as a visual aid while solving the problem. Additionally, you can also try searching for similar problems online to get a better understanding of the concept.

In conclusion, I want to remind you to stay calm and focused, review the relevant principles in your textbook, and seek help if needed. With a clear understanding and application of the principles, I am confident that you will be able to solve this bonus problem and earn the much-needed points. Good luck!

## 1. What is the "Desperate/Bonus problem"?

The "Desperate/Bonus problem" is a mathematical optimization problem that involves finding the minimum or maximum value of a function under certain constraints. It is commonly used in economics, engineering, and other fields to determine the best course of action or decision.

## 2. How is the "Desperate/Bonus problem" different from other optimization problems?

The "Desperate/Bonus problem" is unique in that it allows for both positive and negative values for the dependent variable, whereas most other optimization problems only allow for positive values. This makes it more challenging to solve and requires different techniques and algorithms.

## 3. What are some real-life applications of the "Desperate/Bonus problem"?

The "Desperate/Bonus problem" can be applied to various real-world scenarios, such as determining the optimal production levels for a company, maximizing profits for a business, or finding the best route for a delivery truck. It can also be used in financial planning, resource allocation, and portfolio optimization.

## 4. What techniques are commonly used to solve the "Desperate/Bonus problem"?

There are several techniques that can be used to solve the "Desperate/Bonus problem," including the simplex method, interior point methods, and gradient descent. These methods involve mathematical calculations and iterations to find the optimal solution.

## 5. Is it possible to have multiple solutions for the "Desperate/Bonus problem"?

Yes, it is possible to have multiple solutions for the "Desperate/Bonus problem." This can happen when there are multiple points on the function that satisfy the constraints and have the same minimum or maximum value. In this case, any of these points can be considered a valid solution to the problem.

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