Elephant Rope Friction: Calculating Minimum Force

This problem involves calculating the minimum force required to hold an elephant using a rope wound around a tree trunk. This scenario is a classic application of the Eytelwein's formula, also known as the capstan equation, which deals with friction between a rope (or belt) and a cylindrical surface.

Rope Friction Principles

When a rope is wound around a rough cylinder, the tension on the tight side (the side with the greater force) is significantly larger than the tension on the slack side (the side with the lesser force) due to friction. The amount by which the tension increases or decreases depends on the coefficient of friction and the total angle of wrap of the rope around the cylinder.

The Eytelwein's formula is given by:

\[ F_1 = F_2 e^{\mu \theta} \]

Where:

• \( F_1 \) is the tension on the tight side (the larger force).
• \( F_2 \) is the tension on the slack side (the smaller force).
• \( e \) is Euler's number (approximately 2.71828).
• \( \mu \) (mu) is the coefficient of kinetic friction between the rope and the tree trunk.
• \( \theta \) (theta) is the total angle of wrap of the rope around the trunk, measured in radians.

In our problem, the elephant's pull represents the tight side tension (\( F_1 \)), and the minimum force required to stop the elephant represents the slack side tension (\( F_2 \)). We need to find \( F_2 \).

Given Elephant Problem Parameters

Let's list the known values from the problem statement:

Elephant Force Calculation Steps

To find the minimum force \( F_2 \), we first need to calculate the total angle of wrap \( \theta \) in radians.

1. Calculate the total angle of wrap (\( \theta \)):

   One complete wrap around a circular object is equal to \( 2\pi \) radians.

   Since the rope is wound twice around the tree:

   \[ \theta = \text{Number of wraps} \times 2\pi \text{ radians/wrap} \]

   \[ \theta = 2 \times 2\pi = 4\pi \text{ radians} \]

2. Rearrange Eytelwein's formula to solve for \( F_2 \):

   We have \( F_1 = F_2 e^{\mu \theta} \).

   To find \( F_2 \), we rearrange the formula:

   \[ F_2 = \frac{F_1}{e^{\mu \theta}} \]

3. Substitute the values into the formula and calculate \( F_2 \):

   Now, we substitute the known values of \( F_1 \), \( \mu \), and \( \theta \) into the rearranged formula:

   \[ F_2 = \frac{1000 \text{ kgf}}{e^{(0.3 \times 4\pi)}} \]

   First, calculate the exponent:

   \[ \mu \theta = 0.3 \times 4\pi = 1.2\pi \]

   Using the approximate value of \( \pi \approx 3.14159 \):

   \[ 1.2\pi \approx 1.2 \times 3.14159 \approx 3.76991 \]

   Now, calculate \( e^{\mu \theta} \):

   \[ e^{3.76991} \approx 43.3837 \]

   Finally, calculate \( F_2 \):

   \[ F_2 = \frac{1000 \text{ kgf}}{43.3837} \]

   \[ F_2 \approx 23.0496 \text{ kgf} \]

Final Minimum Force Result

The calculated minimum force required to stop the elephant is approximately \( 23.05 \text{ kgf} \). Comparing this value with the given options, the closest one is 23 kgf.

This demonstrates the significant mechanical advantage provided by wrapping a rope around a rough surface. Even a large force like 1000 kgf from an elephant can be counteracted by a much smaller force with sufficient wraps and friction.