A heavy string of mass 'm' hangs between two fixed points A and B at different levels. The tangents to the string at A and B are at an angle α and β with horizontal as shown in the figure. What is the tension at lowest point of the string?

To find the tension at the lowest point of a heavy string hanging between two fixed points A and B, we need to consider the forces acting on the string and how they relate to the angles α and β at points A and B. The string forms a catenary shape, but for small angles, we can simplify our analysis using basic principles of equilibrium and trigonometry.

Understanding the Forces

When the string is in equilibrium, the forces acting on it must balance out. The weight of the string acts downwards at its center of mass, while the tension in the string acts along its length. At the lowest point of the string, the tension is at its minimum because the weight is distributed along the length of the string.

Analyzing the Angles

The angles α and β represent the inclination of the string at points A and B, respectively. The tension in the string can be resolved into horizontal and vertical components:

• The horizontal component of tension, T_h, is the same at both ends due to equilibrium.
• The vertical components of tension, T_A and T_B, at points A and B can be expressed as:

T_A = T * sin(α) and T_B = T * sin(β)

Setting Up the Equations

At equilibrium, the sum of the vertical forces must equal the weight of the string. If we denote the total weight of the string as W = mg (where g is the acceleration due to gravity), we can write:

T_A + T_B = W

Substituting the expressions for T_A and T_B, we have:

T * sin(α) + T * sin(β) = mg

Finding the Tension at the Lowest Point

Now, we can factor out T from the left side of the equation:

T (sin(α) + sin(β)) = mg

To find the tension T, we rearrange the equation:

T = mg / (sin(α) + sin(β))

Conclusion

This equation gives us the tension at the lowest point of the string in terms of the mass of the string, the gravitational acceleration, and the angles at which the string is inclined at points A and B. It’s important to note that this analysis assumes small angles; for larger angles, a more complex catenary analysis would be necessary. However, for many practical purposes, this formula provides a good approximation.