# Spiraling Inwards: Proving a Particle's Time to Reach a Fixed Point

• ad absurdum
In summary: V^2)}{a^2}} \theta} + e^{-\sqrt{\frac{k(1-V^2)}{a^2}} \theta}}{2}}Now, we can use the definition of the hyperbolic secant function:\text{sech}(x) = \frac{1}{\cosh(x)}Substituting this into our equation for r, we get:r = a \frac{\text{sech}(\sqrt{\frac{k(1-V^2)}{a^2}} \theta)}{2}Finally, we can rewrite this equation in terms of the angle \theta and the time T:r = a \text{sech}(\sqrt
ad absurdum

## Homework Statement

A particle P of mass $$m$$ moves under the infulence of a central force of magnitude $$mkr^{-3}$$ directed towards a fixed point O. Initially $$r=a$$ and P has a velocity $$V$$ perpendicular to OP, where $$V^2 < \frac{k}{a^2}$$. Prove that P spirals in towards O and reaches O in a time

$$T = \frac{a^2}{\sqrt{k-a^2V^2}}$$.

## Homework Equations

$$\frac{d^2u}{d\theta^2} - (\frac{k}{a^2V^2} - 1})u = 0$$

## The Attempt at a Solution

I've got the equation $$r = a sech (\sqrt{\frac{k}{a^2V^2} - 1}) \theta}$$, which I think is right, but I have no idea how to find $T$ from this. I haven't covered hyperbolic functions in much detail before (which is a shame, because they are assumed on this course) so I may be missing something obvious. I'm guessing I should be evaluating some integral but I can't think of anything/see anything useful in my notes. Any hints would be much appreciated.

Thank you for posting your question. It seems like you are on the right track with your equation for r, but you are missing a key step in finding the time T. Let me guide you through the process.

First, let's start with the equation you have for r:

r = a sech (\sqrt{\frac{k}{a^2V^2} - 1}) \theta

We can rewrite this equation as:

r = a \frac{1}{\cosh(\sqrt{\frac{k}{a^2V^2} - 1}) \theta}

Now, let's recall the definition of the hyperbolic cosine function:

\cosh(x) = \frac{e^x + e^{-x}}{2}

Substituting this into our equation for r, we get:

r = a \frac{1}{\frac{e^{\sqrt{\frac{k}{a^2V^2} - 1} \theta} + e^{-\sqrt{\frac{k}{a^2V^2} - 1} \theta}}{2}}

Next, let's use the fact that V^2 < \frac{k}{a^2} to simplify the expression within the square root:

\sqrt{\frac{k}{a^2V^2} - 1} = \sqrt{\frac{k}{a^2} - \frac{k}{a^2}} = \sqrt{\frac{k(1-V^2)}{a^2}}

Substituting this back into our equation for r, we get:

r = a \frac{1}{\frac{e^{\sqrt{\frac{k(1-V^2)}{a^2}} \theta} + e^{-\sqrt{\frac{k(1-V^2)}{a^2}} \theta}}{2}}

Now, let's use the fact that V^2 < \frac{k}{a^2} to simplify the expression within the square root:

\sqrt{\frac{k}{a^2V^2} - 1} = \sqrt{\frac{k}{a^2} - \frac{k}{a^2}} = \sqrt{\frac{k(1-V^2)}{a^2}}

Substituting this back into our equation for r, we get:

r = a \frac{1}{\frac{e^{\sqrt{\frac{k

I would first commend the student for their attempt at a solution and for recognizing their limitations in regards to the use of hyperbolic functions. I would suggest that they review their notes on hyperbolic functions and their properties, as well as the relevant equations for motion under a central force. It may also be helpful to consult with a classmate or the instructor for clarification and guidance on how to approach this problem.

In general, when solving for the time it takes for a particle to reach a fixed point, we can use the equation T = \frac{1}{2\pi} \sqrt{\frac{m}{k}} \int_{r_1}^{r_2} \frac{dr}{r^2\sqrt{\frac{2}{m}(E-U(r))}}, where E is the total energy of the particle and U(r) is the potential energy function. In this case, the potential energy function is given by U(r) = \frac{1}{2}mkr^{-2}, so we can substitute this into the equation and solve for T. Alternatively, we can use the equation T = \frac{2\pi}{\sqrt{k}} \sqrt{\frac{m}{2}} \int_{r_1}^{r_2} \frac{dr}{\sqrt{E-U(r)}}.

I would also remind the student to carefully consider the initial conditions and to check their solution for any mistakes or missing steps. It may also be helpful to draw a diagram or plot the trajectory of the particle to gain a better understanding of the problem. With persistence and a thorough understanding of the relevant equations, the student should be able to successfully solve for T and confirm the expected result that the particle spirals in towards the fixed point O and reaches it in a finite time.

## 1. What is the purpose of "Spiraling Inwards" in particle research?

The concept of "Spiraling Inwards" is used to understand the behavior of particles as they move towards a fixed point. This is important in particle research because it helps us understand the time it takes for particles to reach a specific destination, which can have implications in various fields such as physics, chemistry, and engineering.

## 2. What is a fixed point in particle research?

A fixed point in particle research refers to a specific location or point in space that a particle is moving towards. This can be a physical object or a theoretical point used for calculations. The concept of "Spiraling Inwards" is used to determine the time it takes for a particle to reach this fixed point.

## 3. How is the time to reach a fixed point calculated in "Spiraling Inwards"?

The time to reach a fixed point in "Spiraling Inwards" is calculated using mathematical equations and principles such as Newton's laws of motion and calculus. The particle's initial velocity, acceleration, and the distance to the fixed point are taken into account to determine the time it takes for the particle to reach its destination.

## 4. What are the applications of "Spiraling Inwards" in real-world scenarios?

The concept of "Spiraling Inwards" has various applications in real-world scenarios such as determining the time it takes for a rocket to reach its target in space, calculating the trajectory of a projectile, and predicting the time it takes for a chemical reaction to occur. It also has implications in fields such as astrophysics and nuclear physics.

## 5. How is "Spiraling Inwards" relevant in understanding the behavior of subatomic particles?

Subatomic particles, such as electrons, follow the laws of motion and can be affected by forces such as gravity, electric and magnetic fields. Understanding the time it takes for these particles to reach a fixed point is important in studying their behavior and interactions. "Spiraling Inwards" provides a framework for analyzing the movement of subatomic particles towards a fixed point, giving us insights into their properties and behavior.

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