Tutorial Sheet: Statics III- Common Catenary
1. Derive the intrinsic equation of common catenary.
2. If T be tension at any point P of a catenary, and 𝑇𝑇0 that at the lowest point A, prove that
𝑇𝑇 2 − 𝑇𝑇02 = 𝑊𝑊 2 , W being eight of the arc AP of the catenary.
𝑦𝑦+𝑠𝑠
3. Show that for a common catenary 𝑥𝑥 = 𝑐𝑐 log( ).
𝑐𝑐
4. A rope of length 2l feet is suspended between two points at the same level, and the
lowest point of the rope is b feet below the points of suspension. Show that the horizontal
component of the tension is (𝑙𝑙 2 − 𝑏𝑏 2 )/2𝑏𝑏 , w being the weight of the rope per foot of its
length.
5. A uniform chain of length l, which can just bear a tension of n times its weight, is
stretched between two points on the same horizontal line. Show that the least possible
1
sag in the middle is 𝑙𝑙 �𝑛𝑛 − �𝑛𝑛2 − �.
4
6. A given length, 2s, of a uniform chain has to be hung between two points at the same
level and the tension has not to exceed the weight of a length b of the chain. Show that
𝑏𝑏+𝑠𝑠
the greatest span is √𝑏𝑏 2 − 𝑠𝑠 2 log .
𝑏𝑏−𝑠𝑠
7. The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the
1+�1+𝜇𝜇2
ratio of maximum span to the length of the chain is 𝜇𝜇 log � � where 𝜇𝜇 is the
𝜇𝜇
coefficient of friction.
8. A weight W is suspended from a fixed point by a uniform string of length l and weight w
per unit length. It is drawn aside by a horizontal force P. Show that in the position of
equilibrium, the distance of W from the vertical through the fixed point is
𝑃𝑃 𝑊𝑊+𝑙𝑙𝑙𝑙 𝑊𝑊
𝑤𝑤
�sinh−1 � 𝑃𝑃 � − 𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 � ��.
𝑃𝑃
9. A uniform chain, of length l and weight W, hangs between two fixed points at the same
level, and a weight 𝑊𝑊′ is attached at the middle point. If k be the sag in the middle, prove
𝑘𝑘 𝑙𝑙 𝑙𝑙
that the pull on either point of support is 𝑊𝑊 + 𝑊𝑊 ′ + 𝑊𝑊.
2𝑙𝑙 4𝑘𝑘 8𝑘𝑘
By Avinash Singh, Ex IES, B.Tech IIT Roorkee 1|Page
10. A heavy string of uniform density and thickness is suspended from two given points in
the same horizontal plane. A weight, an nth that of the string, is attached to its lowest
point. Show that if 𝜃𝜃, 𝜑𝜑 be the inclinations to the vertical of the tangents at the highest
and lowest points of the string tan 𝜑𝜑 = (1 + 𝑛𝑛) tan 𝜃𝜃
11. Find the length of an endless chain which will hang over a circular pulley of radius a so
3 4𝜋𝜋
as to be in contact with two third of the circumference of the pulley.�𝑎𝑎 � + ��
𝑙𝑙𝑙𝑙𝑙𝑙(2+√3 3
By Avinash Singh, Ex IES, B.Tech IIT Roorkee 2|Page
