A cable of length L+Δis suspended between two points that...

A cable of length $L+\Delta$ is suspended between two points that are at the same elevation and a distance $L$ apart. ( $a$ ) Assuming that $\Delta$ is small compared to $L$ and that the cable is parabolic, determine the approximate sag in terms of $L$ and $\Delta .(b)$ If $L=100 \mathrm{ft}$ and $\Delta=4 \mathrm{ft}$, determine the approximate sag. [Hint: Use only the first two terms of Eq. $(7.10) .]$

A cable of length $L+\Delta$ is suspended between two points that are
at the same elevation and a distance $L$ apart. ( $a$ ) Assuming that $\Delta$ is
small compared to $L$ and that the cable is parabolic, determine the
approximate sag in terms of $L$ and $\Delta .(b)$ If $L=100 \mathrm{ft}$ and $\Delta=4 \mathrm{ft}$, determine the approximate sag. [Hint: Use only the first two terms of Eq. $(7.10) .]$

Key Concepts

Sag

Sag is the term used to describe the maximum vertical displacement of the cable, typically occurring at the midpoint between the supports. In problems involving suspended cables, determining the sag is crucial as it relates the inherent flexibility of the cable to the additional length beyond the horizontal distance between the supports, indicating how much the cable devotes to curving downward under load.

Small-Parameter Approximation

This concept arises when one or more parameters (such as the extra length of the cable compared to the span) are small relative to the main dimensions of the system. In this context, assuming a small extra length allows for approximations like Taylor series expansion or considering only the leading terms in the resulting equations, thus significantly simplifying the calculations involved in determining the sag.

Parabolic Cable Model

This concept involves approximating the shape of a hanging cable as a parabola, which is valid under certain conditions such as when the sag (the vertical displacement at the midpoint) is small relative to the span. The parabolic model simplifies the analysis by reducing the problem from the more complex catenary curve to a quadratic function, making it easier to relate the cable’s extra length to its sag.

Transcript

00:02 Hi, so in this problem that i'm given to solve it is given that it is given that between two points a cable is suspended it is the cable is suspended and the cable is the shape of the cable is parabolic and these these two points at the same horizontal distance and they are distance l apart this distance between the two points from which the cable is suspended is l it is also even that it is also given that the let the total length of this cable is l plus delta usually denote this length of the level by s so you can see that s is equal to l plus delta and it's also given that delta is very very small compared to l okay so we know that the lowest point of the cable the distance from this horizontal from this horizontal line to this lowest point of the cable this is the sag and this sag is this is noted by h so our question the part a of our question is that we need to determine h in h in terms of l and delta this is our question so we know that if we take the lowest point of the cable as the origin and draw an xy coordinate from this point this is the x coordinate and this is the y coordinate so starting from here and going to any point in the cable we can get the starting from here and going to any point of the cable we can get the length of the cable starting from the origin up to that point so the point you will consider here is this point this second point from which the cable is suspended and our formula for for getting the length of the cable up to this point so our formula for getting the length of the cable up to this point let's consider this length this length from the origin to the second insulator as s -1 so we know that if if this length is s1 then s is our total length is equal is twice of this s one and according to our formula we know we know that s1 is equal to let me add the formula first so the formula so according to the formula s 1 is equal to x into 1 plus 2 over 3 s into 1 plus 2 over 3 and y over x whole square so we've it is given in the question that we have to take only two terms of this equation so this is enough enough for our use okay so this implies so this x is the x coordinate of this…

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