Capstan Equation Calculator

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Enter values of tension, coefficient, wrap angle etc to use our basic and advanced Capstan Equation Calculator online!

Welcome to the Capstan Equation Calculator! Capstans, commonly used in sailing, engineering, and various industrial applications, rely on the Capstan Equation to understand the relationship between tension, friction, and wrap angle.

Formula:

The formula is:

$T_2 = T_1 \times e^{(\mu \theta)}$

Variable	Meaning
$T_2$	Tension on the loaded side of the rope
$T_1$	Tension on the free side of the rope
$\mu$	Coefficient of friction between the rope and the capstan surface
$\theta$	Angle of contact (in radians) between the rope and the capstan
$e$	Euler’s number (approximately 2.71828)

How to Calculate ?

In order to calculate the tension on the loaded side of the rope ( $T_2$ ) using the Capstan equation, first, determine the tension on the free side ( $T_1$ ), the coefficient of friction ( $\mu$ ) between the rope and the capstan, and the angle of contact ( $\theta$ ) in radians. Then, multiply the tension on the free side ( $T_1$ ) by Euler’s number $e$ raised to the power of the product of $\mu$ and $\theta$ .

Solved Calculations :

Example 1:

Given:

$T_1$ = 100 N
$\mu$ = 0.3
$\theta$ = $\pi$ radians (180 degrees)

Calculation	Instructions
Step 1: $T_2 = T_1 \times e^{(\mu \theta)}$	Start with the formula.
Step 2: $T_2 = 100 \times e^{(0.3 \times \pi)}$	Replace $T_1$ , $\mu$ , and $\theta$ with their values.
Step 3: $T_2 = 100 \times e^{0.94248}$	Multiply $\mu$ by $\theta$ .
Step 4: $T_2 = 100 \times 2.566$	Calculate $e^{0.94248}$ using Euler’s number.
Step 5: $T_2 = 256.6 \, \text{N}$	Multiply by the tension on the free side.

Answer:
The tension on the loaded side ( $T_2$ ) is 256.6 N.

Example 2:

Given:

$T_1$ = 150 N
$\mu$ = 0.5
$\theta$ = $\frac{\pi}{2}$ radians (90 degrees)

Calculation	Instructions
Step 1: $T_2 = T_1 \times e^{(\mu \theta)}$	Start with the formula.
Step 2: $T_2 = 150 \times e^{(0.5 \times \frac{\pi}{2})}$	Replace $T_1$ , $\mu$ , and $\theta$ with their values.
Step 3: $T_2 = 150 \times e^{0.7854}$	Multiply $\mu$ by $\theta$ .
Step 4: $T_2 = 150 \times 2.193$	Calculate $e^{0.7854}$
Step 5: $T_2 = 328.95 \, \text{N}$	Multiply by the tension on the free side.

Answer:
The tension on the loaded side ( $T_2$ ) is 328.95 N.

What is Capstan Equation ?

The Capstan Equation Calculator uses the capstan equation to calculate the tension in a rope or cable wrapped around a cylindrical object like a capstan, pulley, or drum. This equation is critical in physics and engineering, especially in systems where friction plays a vital role in transmitting force, such as winches, hoists, and cranes.

Final Words:

The Capstan Equation Calculator helps calculate the tension in a rope or cable wrapped around a cylinder, like a capstan or pulley. The engineers and mechanics working with tension-based systems, such as winches and cranes can take benefit from this greatly!

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