A cable subjected to its own weight and free of any other loads will take the form of
Catenary curve
Understanding Cable Shapes Under Weight
The question asks about the specific shape a cable takes when it is hanging freely, supported at its ends, and the only load acting upon it is its own weight, which is distributed uniformly along the length of the cable.
Shape of a Cable Under its Own Weight
When a flexible cable is subjected only to its self-weight, the shape it forms is known as a catenary curve. This is a fundamental concept in physics and engineering related to static equilibrium.
Let's consider why this is the case:
- A cable is a flexible structure, meaning it can only resist tension, not compression or bending.
- When hanging freely under its own weight, the tension forces within the cable adjust at every point to balance the gravitational force acting on the cable segment below that point.
- The mathematical description of this equilibrium configuration results in the equation of a catenary.
Analyzing the Options
Let's look at the other options to understand why they are not correct for a cable hanging under its own weight:
- Parabolic curve: A parabolic shape is formed by a cable when it is subjected to a load that is uniformly distributed horizontally (like the weight of a bridge deck supported by hangers from the cable), not uniformly distributed along the length of the cable itself. The equations describing the two load distributions lead to different curve shapes.
- Elliptic curve: An ellipse is a closed curve, typically associated with orbits or intersections of planes with cones. It is not the shape formed by a hanging cable under gravity.
- Bernoulli's Lemniscate: This is a specific algebraic curve with a shape resembling an "infinity" symbol (∞). It arises in various mathematical contexts but is not related to the shape of a hanging cable.
Therefore, the unique shape formed by a cable hanging under its own uniformly distributed weight along its length is the catenary curve.
Mathematical Representation of a Catenary
The equation of a catenary can be expressed as:
\( y = a \cosh\left(\frac{x}{a}\right) \)
where:
- \( y \) is the vertical position.
- \( x \) is the horizontal position.
- \( a \) is a parameter related to the tension at the lowest point of the cable and the weight per unit length.
- \( \cosh \) is the hyperbolic cosine function.
This mathematical form confirms that the catenary is distinct from a parabola, ellipse, or lemniscate.
| Curve Type | Typical Loading Scenario for Cables |
|---|---|
| Catenary | Cable subjected only to its own weight (uniform load along length). |
| Parabola | Cable subjected to a uniformly distributed horizontal load (e.g., bridge deck). |
| Ellipse | Not applicable to hanging cables under gravity. |
| Bernoulli's Lemniscate | Not applicable to hanging cables under gravity. |
In summary, the shape of a cable under its own weight is a catenary curve due to the nature of the gravitational forces acting uniformly along its length and the cable's flexibility.
Revision Table: Cable Shapes
| Shape | Condition |
|---|---|
| Catenary | Cable hanging freely under its own weight (uniform load along length). |
| Parabola | Cable supporting a uniformly distributed horizontal load. |
Additional Information: Catenary Properties and Applications
The term "catenary" comes from the Latin word "catena," meaning chain. Early mathematicians like Galileo initially thought the curve was a parabola, but later, Bernoulli, Leibniz, and Huygens proved it was a distinct curve, the catenary.
Applications of the catenary shape include:
- Design of suspension bridge cables (though often simplified to a parabola due to the dominance of deck load).
- Overhead power lines.
- Chains hanging between two points.
- Certain architectural designs (e.g., St. Louis Arch, inverted catenary).
The catenary is the shape that minimizes potential energy for a hanging chain or cable under gravity, making it a stable equilibrium configuration.
Important Questions from Cables and Frames
If a beam supports two concentrated loads, then the shape of profile followed by cable is:
The lateral deflection of a frame is called as__________.
Frames are characterized by __________ resisting members at some or all the joints.
In which of the following supports of frames, both reactions are always vertical and may be found out by the principle of moments?
A suspension cable, supporting loads, will be under:
