Question

A heavy string hangs over two fixed small smooth pegs. The two ends of the string
are free and the central portion hangs in a catenary. Show that the free ends of the
string are on the directrix of the catenary. If the two pegs are on the same level and
distant 2a apart, show that the equilibrium is impossible unless the string is equal
to or greater than 2ae.
​

Answer 1

The free ends of the string hang on the directrix of the catenary curve, and for equilibrium to be possible with two pegs on the same level and distant 2a apart, the length of the string should be equal to or greater than 2ae.

Explanation:

• To show that the free ends of the string are on the directrix of the catenary, we can consider the properties of a catenary curve. A catenary is the curve formed by a hanging flexible chain or cable under its own weight.
• In a catenary, the curve is symmetric and the directrix is a horizontal line passing through the lowest point of the curve. The directrix is the line of symmetry for the catenary.
• In the given scenario, the string hangs over two fixed small smooth pegs, which means that the string is under tension and forms a catenary. Since the string is heavy, it will take the shape of a catenary curve.
• As the two ends of the string are free, they can only lie on the directrix of the catenary curve because the directrix is the line of symmetry and the lowest point of the curve. Therefore, it can be concluded that the free ends of the string are on the directrix of the catenary.
• Now, let's consider the equilibrium of the system. If the two pegs are on the same level and distant 2a apart, the equilibrium is only possible if the length of the string is equal to or greater than 2ae, where e is the eccentricity of the catenary curve.
• The reason for this is that the tension in the string is responsible for supporting the weight of the string itself. For the equilibrium to be maintained, the tension at each end of the string should be sufficient to balance the weight of the string in between. If the string is shorter than 2ae, the tension will not be enough to support the weight, and the equilibrium will be impossible.
• In conclusion, the free ends of the string hang on the directrix of the catenary curve, and for equilibrium to be possible with two pegs on the same level and distant 2a apart, the length of the string should be equal to or greater than 2ae.

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