# Exploring a Captivating Problem in Classical Mechanics

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## Chapter 1: Introduction to the Problem

Classical mechanics is a remarkable and refined method for illustrating real-world motion through mathematical concepts. Recently, I encountered a challenging examination question that initially appeared daunting. However, after thoroughly analyzing the situation mathematically, I found the solution to be immensely satisfying—so much so that I felt compelled to share it with you.

### The Problem: A String Winding Around a Pole

Consider a heavy particle resting on a smooth horizontal table, tethered to one end of a light, inextensible string of length L. The other end of the string is affixed to a point P on the edge of a vertical pole, which is anchored to the table. The base of the pole forms a circular area with a radius a, centered at point O on the table.

At the beginning, at time t = 0, the string is taut and oriented perpendicularly to the line OP. Upon striking the particle, it begins to wind the string around the pole while remaining taut, meaning that, at a later time t, a segment of the string contacts the pole.

If the particle's initial velocity is v, we aim to demonstrate that it will collide with the pole at a time given by L²/(2av).

### Solution Part 1: Understanding the Situation

This may seem complex at first glance, but like many mechanics problems, visualizing the scenario can help clarify the approach. Most of the introductory details remind us that we're working within two dimensions since the mass rests on the table and is connected to a point on the table, allowing it to move solely in that plane.

Here’s a sketch I created to illustrate this problem, which I will explain further:

In my first sketch, as described, the entire length of the string is positioned at a right angle to the radius of the pole before any movement occurs. Once motion begins, as shown in the second sketch, the string starts to wrap around the pole. It's crucial to note that the tension in the string always acts perpendicularly to the particle’s movement, leading to a constant angular velocity, assuming no other forces intervene.

The length of the string in contact with the pole will depend on the angle θ that the string subtends at the pole. With θ expressed in radians, the length of the string touching the pole can be calculated as aθ. This results in a remaining length of L - aθ that is not in contact with the pole. As time progresses, θ will increase, making it a function of time—θ(t).

To find the point at which our particle will hit the pole, we can draw several right-angled triangles in our second diagram (highlighted in red). By applying basic trigonometry, we can determine the x and y coordinates of the particle relative to point O.

For the x-coordinate, we sum the two horizontal dashed lines, represented as asinθ + (L - aθ)cosθ. For the y-coordinate, we move upward by acosθ and then downward by (L - aθ)sinθ, resulting in acosθ - (L - aθ)sinθ.

### Solution Part 2: Deriving Velocity in Terms of θ

Now that we have the x and y coordinates of our particle in relation to point O, we need to calculate the particle's velocity to determine the time until it hits the pole. Velocity is defined as the derivative of position concerning time, so we can differentiate our position coordinates to find the velocity coordinates of the particle. Remembering that θ is a function of time, we will denote the derivative of a function with a dot above it.

Using the chain and product rules of differentiation, we can derive the horizontal component of the particle’s velocity:

Similarly, applying this method to the y position coordinate gives us the vertical component of the particle’s velocity:

Now, using Pythagoras’ Theorem on the horizontal and vertical components of the particle’s velocity, we can derive the overall speed of the particle in terms of θ:

This simplifies using the trigonometric identity sin²θ + cos²θ = 1, along with (a - b)² = (b - a)².

### Final Part: Calculating the Time Until Impact with the Pole

We now have an expression for the particle's speed in terms of θ. We know the particle starts with an initial velocity v, and since the only other force in play after the particle begins to move is the tension in the string, we can assume it maintains a constant angular velocity. Thus, the distance traveled by the particle can be expressed as vt.

By integrating our speed concerning t, we can determine the distance traveled, which is expected to equal vt:

Let’s introduce a constant of integration, c, but since we know that θ = 0 at t = 0 (prior to any movement), we find that c = 0, leading to:

To find the moment when the particle collides with the pole, we need to determine when the angle subtended by the string equals the full length of the string, which occurs when θ = L/a. Substituting this into our final equation provides us with:

What do you think about this solution? Feel free to share your thoughts!