A high-voltage power line spans a 40-meter horizontal distance between two transmission towers, resulting in a 10-meter vertical sag due to the effects of gravity and thermal expansion. The curve formed by the suspended cable is a catenary, which accurately models the behavior of a uniform, flexible cable under its own weight. Unlike a parabolic shape, the catenary is described by the hyperbolic cosine function and offers a precise representation of the cable's form.
In this setup, engineers assign the parameter aaa a value of 20 meters. This parameter corresponds to the ratio between the horizontal tension in the cable and its weight per unit length. By placing the origin at the lowest point of the curve, the towers are located symmetrically at positions 20 meters to either side of the curve.
To determine the exact cable length required for installation, the arc length of the catenary between the two towers is calculated. The formula used is:
The result of this integration is:
This value gives the total length of the power line needed, ensuring proper installation while accounting for the sag caused by the cable’s own weight and thermal effects.
A high-voltage power line hanging between two transmission towers spans a 40-meter gap with a significant 10-meter vertical sag due to thermal expansion.
The objective is to find the exact length of the power line required to connect these points safely.
Engineers model this using a catenary, the natural curve of a hanging cable described mathematically by the hyperbolic cosine function.
This curve is controlled by a parameter a, which relates the cable’s weight per meter to the horizontal tension. For this specific installation, the value of parameter a is assumed to be 20 meters.
To find the exact length, the calculation uses the arc length function. Here, the derivative of the catenary gives the hyperbolic sine.
A standard identity involving hyperbolic functions simplifies the expression further by removing the square root.
The integral of the resulting hyperbolic cosine is simply the hyperbolic sine. Evaluating the hyperbolic sine function at the limits from negative twenty to positive twenty gives a total length of approximately 47 meters.
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