Show that the curve assumed by a cable that carries a...

Show that the curve assumed by a cable that carries a distributed
load $w(x)$ is defined by the differential equation $d^{2} y / d x^{2}=w(x) / T_{0}$
where $T_{0}$ is the tension at the lowest point.

Key Concepts

Static Equilibrium

Static equilibrium is fundamental to analyzing systems under load. In this context, the cable is subjected to forces that must balance in both the horizontal and vertical directions, ensuring that the net force and moment on any segment are zero. This balance leads to the derivation of the governing differential equations that describe the cable’s shape.

Distributed Load

A distributed load is one that acts continuously along the length of the cable, typically expressed as a function w(x) representing the force per unit length at any point. This continuous load influences the curvature of the cable by contributing to the net vertical force acting on segments of the cable.

Second-Order Differential Equation

The second-order differential equation, d²y/dx² = w(x)/T?, describes how the curvature of the cable (the second derivative of its deflection) is related directly to the distributed load and inversely to the horizontal tension T? at the lowest point. This mathematical expression encapsulates the physical relationship between force distribution and deflection in a hanging cable.

Tension in Cables

The tension in the cable, particularly the horizontal component at its lowest point (T?), plays a critical role in determining the cable's response to loading. This tension counteracts the applied distributed load, and its constancy along the cable (under certain assumptions) simplifies the derivation of the cable's curvature.

Transcript

00:03 Hi.

00:05 So in this problem it is given that we have a cable that is suspended and the load acting on it is a function of the distance from the lowest point and the tension acting on the lowest point of the cable is t -not and we have to prove that the shape of the cable is given by the double derivative of the double derivative of y is equal to w x over t0 and is given that the if if suppose this is the cable then this is the cable then the cable tension at the lowest point is t not in this height of any point from the origin like if the x and y coordinates are these x and y then then this height of any point is is given by y and the distance from the lowest point is given by x okay so we have to prove this thing okay so let's so for this let's take an element of the cable not the complete cable but a little part of the cables that we can apply differentiation equations to to that to that part so so i take a part of the cable a segment of the cable let this be the segment and we know that a tension is acting on the cable in the horizontal direction t not and simply there will be horizontal tension here t not if if this point is the origin then not not the origin any any point at with the the vertical tension is a function of x if x is the coordinate of this point and if this distance is delta x then the tension here is t y x plus delta x because that's the coordinate of this point the x coordinate of this point and and this distance is delta y like the where it is this is element of the wire small segment of the wires if this point is at elevation zero this would be elevation delta y so this would be delta y higher than this point and this would be delta x ahead than this point this point okay so let's let's solve for let's solve for this and we know that the the cable is in an equilibrium so all the forces present in the wire direction should cancel out each other.

03:46 It's a submission of the forces and the y direction should be equal to zero...

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