To solve the problem of how much thread has passed through the boy's hand when the cylinder reaches him, we can break it down into a few logical steps. We need to consider the relationship between the distance the cylinder travels and the amount of thread unwound from it. Let's go through this step by step.
Understanding the Cylinder's Motion
When the boy pulls the thread with a force F, the cylinder will start to roll towards him. The key here is to recognize that as the cylinder rolls, the thread unwinds from it. The distance the cylinder travels is directly related to the length of the thread that has passed through the boy's hand.
Distance Traveled by the Cylinder
The cylinder starts at a distance l from the boy. As the boy pulls the thread, the cylinder rolls towards him. When the cylinder reaches the boy, it has traveled a distance equal to l.
Relating Distance to Thread Length
As the cylinder rolls, the amount of thread that unwinds is equal to the distance the cylinder travels multiplied by the number of times the cylinder rotates. The circumference of the cylinder is given by the formula:
For every complete rotation of the cylinder, a length of thread equal to the circumference unwinds. Therefore, if the cylinder rolls a distance l, the number of rotations (N) it makes can be calculated as:
- N = Distance traveled / Circumference = l / (2πr)
Calculating the Length of the Thread
Now, to find the total length of the thread that has passed through the boy's hand, we can multiply the number of rotations by the circumference:
- Length of thread (L) = N × C = (l / (2πr)) × (2πr)
Notice that the 2πr terms cancel out, simplifying our equation to:
Final Result
Thus, the length of the thread that has passed through the boy's hand when the cylinder reaches him is simply equal to the initial distance l. In conclusion, if the cylinder rolls a distance of l towards the boy, the length of the thread that has unwound and passed through his hand is also l.
