An electrical cable is atteched to the electrical pole 25m apart. The cable weighs 2.5N/m and can resist a pull of 300N . assuming that the cable takes a parabolic shape, detrmine the smallest value of sag in the cable
To determine the smallest value of sag in the cable, we need to find the point of maximum deflection, where the cable takes a parabolic shape.
Given:
Distance between the electrical pole = 25 m
Weight of the cable = 2.5 N/m
Pull resistance of the cable = 300 N
Let's assume that the point of maximum deflection occurs at a height 'h' above the midpoint of the cable.
1. Firstly, let's find the weight of the entire cable. Since the weight of the cable is given as 2.5 N/m, we can calculate it as:
Weight of the cable = 2.5 N/m * 25 m = 62.5 N
2. Now, we need to find the point of maximum deflection. This occurs at the midpoint of the cable. So, the length of the cable on either side of the midpoint is:
Length on either side = 25 m / 2 = 12.5 m
3. We can calculate the tension at the midpoint by balancing the weight of the cable and the pull resistance:
Tension at the midpoint = Weight of the cable / 2 = 62.5 N / 2 = 31.25 N
4. Using the concept of a parabolic shape, we know that the tension at any point on the cable can be calculated using the equation:
Tension = Tension at the midpoint + Weight of the cable * (distance from the midpoint)^2 / (2 * Length on either side)^2
Let's substitute the known values into the equation:
300 N = 31.25 N + 62.5 N * (h)^2 / (2 * 12.5 m)^2
5. Simplifying the equation:
268.75 N = 1562.5 N * h^2 / 312.5 m^2
6. Rearranging the equation to solve for 'h^2':
h^2 = (268.75 N * 312.5 m^2) / 1562.5 N
h^2 = 53.75 m^2
7. Taking the square root of both sides to get the value of 'h':
h = √(53.75 m^2)
h ≈ 7.34 m
Therefore, the smallest value of sag in the cable is approximately 7.34 meters.