Equilibrium of Rigid Bodies (A/L Combined Mathematics)

1. A uniform rod of weight W, is hanging at equilibrium with two strings connected at the ends of the rod. If the tensions of the strings are T1, T2, show that the cosine of the angle of inclination of the rod to the vertical is,

|T12−T 2

2|W √2 (T 1

2+T22 )−W 2

. (A.L. 1983)

2. A rod AB of weight w whose centre of gravity divide it in to two portions of length a and b has been tied a light inextensible string of length l (> a +b) to its two ends. The string is slung over a small smooth peg P and the rod is in equilibrium.

a) Show that,i) AP̂G = BP̂G

ii) cos AP̂G = a+b2 l [ l2−(a+b )2

ab ]12

b) Find the tension in the string. (A.L. 1991)

3. A rod AB of weight W rests wholly within a fixed smooth hemispherical bowl of radius r and centre C. The centre of gravity G of AB divides it into two portions of length a and b where b>a and r>√ab . If 𝜃 is the inclination of the rod to the horizontal in the equilibrium position, show that,

sin θ= b−a

2√r2−ab and CG = √r2−ab

Find the reactions between the rod and the bowl. (A.L. 1992)

4. A uniform solid sphere of weight w and radius a, is hung by a string of length a from a fixed point O. A uniform rod, also of weight w and length 4a has one end freely attached to the same [point o. If the rod rests

touching the sphere, show that the inclination of the string and the rod to the vertical are each equal to π12 .

show also that the tension in the string is w cos π

12

sin π3

and find the reaction between the sphere and the rod.

(A.L. 1993)

5. A uniform smooth rod AB of length 2a and weight w can turn freely about its fixed end A. A small smooth ring C of weight 2w can slide along the rod. The ring is jointed to a fixed point D, in the same horizontal

level as the point A by an inextensible string of length a4 . In the position of equilibrium, find the reaction

between the rod and the ring and show that the rod makes an angle π3 with the horizontal. Also find the

tension in the string and the reaction at the end A. (A.L. 1994)

6. If three non- parallel coplanar forces acting upon a rigid body keeping it in equilibrium, show that they must meet at the same point.

Two uniform smooth spheres of different radii with centers A and B and having equal weight w rest inside a fixed smooth right circular hollow cone which has its vertex downwards and each sphere touches the cone

at one point only. The semi-vertical angle of the cone is π3 and its axis makes an angle 𝛽 ( <

π6 with the

vertical. If the line AB makes an angle 𝜃 with the upward vertical, show that.𝜃 = tan−1(cot 2 β−12

cosec 2β ) .Find the reactions of the sides of the cone. (A.L. 1997)

(P.T.O.) 7. . If three non- parallel coplanar forces acting upon a rigid body keeping it in equilibrium, show that they must meet at the same point.

A smooth uniform hemispherical bowl, of radius r, rests on smooth horizontal table and partly inside it rests a smooth uniform rod, of length 2l and of weight equal to that of the bowl. In the position of equilibrium, the

inclination of the base of the hemisphere to the horizontal is 𝛼 ( π2 ) and the angle subtended at the centre by

the part of the rod within the bowl is 2𝛽 (¿ π2 ). Show that ,

i) R = l cosec 𝛼 sin (𝛼 + 𝛽)

ii) Cot 𝛼 = tan 2𝛽 - 12 sec 2𝛽.

(A.L.1998)

8. A smooth peg is fixed at a point P at distance a from a smooth vertical wall. A uniform rod AB of length 6a and weight w is in equilibrium resting on the peg with the end A in contact with the wall.

Taking 𝜃 to be the angle made by the rod AB with the horizontal draw a triangle of forces, representing forces acting on the rod. Find the reaction at P, in terms of w and 𝜃. Show that,3 cos2θ = 1. (A.L. – 2001)

9. Five equal uniform rods, each of weight W are hinged freely at their ends to form a regular pentagon ABCDE. The pentagon is placed in a vertical plane with CD resting on a horizontal plane and the regular pentagonal form is maintained by means of a light rod connecting the mid points of BC and DE. Indicate the forces acting on the rods AB and BC.

Also, prove tat the tension in the light rod is (cot π4+3cot 2π

5 )W . (A.L. 2003)

10. AB, BC and CD are three uniform rods of equal weight and length, smoothly hinged at B and C. The ends A and D are hinged to fixed smooth horizontal pins at the same level. The system hangs in equilibrium. If AB and CD are inclined at the same angle 𝛼 to the horizontal, and 𝛽 is the inclination of the reaction at A

on AB to the horizontal, show that tan𝛼 = 23 tan𝛽. (A.L.

2004)

11. Two smooth uniform rods AB, BC each of equal length 2a and weight W, are freely hinged at B, and are suspended by two light inextensible strings AO, CO each of length 2a, tied to a fixed point O. A uniform

sphere of weight W and radius a3 rests in contact with the rods and is suspended by them. Show that, in

position of equilibrium, each rod makes with the vertical an angle 𝜃 given by cot3θ + cot θ -30 = 0.

Find the only possible value of cot θ, and hence, show that the reaction at the hinge B is W. (A.L. 2009)

12. Two uniform rods AB and BC are equal in length. The weight of AB is 2w and the weight of BC is w. The rods are smoothly hinged at B and the mid points of the rods are connected by a light inelastic string. The system stands in equilibrium in a vertical plane with A and C on a smooth horizontal table.

If AB̂C= 2𝜃, show that the tension of the string is 32w tan𝜃.Find the magnitude of the reaction at B and the angle it makes with the horizontal. (A.L. 2011) 13.Two uniform smooth spheres, each of weight w and radius b, rest inside a hollow cylinder of radius a (< 2b), fixed with its base horizontal. Show that the reaction between the curved surface of the cylinder

and each sphere is (a−b ) w

√2ab−a2, and find the reactions between the two spheres.

W.M.J.P.Wanigasekera (B.Sc./PG Dip/M.Sc.)