How to derive the expression for tension in a simply supported transmission line and calculate it for specific values?
The expression for tension in a simply supported transmission line of length 1 and density p for fundamental frequency f1 is given by T = (pi^2)*p*f1^2. When the values of length (l), linear density (p), and fundamental frequency (f1) are known, the tension can be calculated using this formula.
Given values: l = 20 m, p = 5 kg/m, f1 = 15 Hz
Substitute these values into the formula: T = (pi^2 * p * l * f1^2)/4
T = (pi^2 * 5 * 20 * 15^2)/4 = 4428 N
Therefore, at l = 20m, p = 5kg/m, and f1 = 15Hz, the tension required in the simply supported transmission line is 4428 N.
Understanding the Expression for Tension
Tension Formula: T = (pi^2 * p * l * f1^2)/4
When a transmission line is modeled as a string and supported at both ends, the tension required for it to vibrate at its fundamental frequency can be calculated using the above formula. The tension (T) depends on the linear density (p) of the string, the length of the string (l), and the fundamental frequency (f1) of the vibration.
In the given scenario with l = 20m, p = 5kg/m, and f1 = 15Hz, we can see that the tension required is 4428 N. This tension value ensures that the transmission line vibrates at its fundamental frequency of 15Hz.
This calculation is important in engineering applications where understanding the relationship between tension, length, density, and frequency is crucial for designing and maintaining transmission lines to operate at desired frequencies.