Engineering Mechanics

(^68) „„„„„ A Textbook of Engineering Mechanics
Example 5.9. A uniform rod AB of length 3r remains in equilibrium on a hemispherical
bowl of radius r as shown in Fig. 5.21.
Fig. 5.21.
Ignoring friction find the inclination of the rod (θ) with the horizontal.
Solution. Given : Length of the rod AB = 3r and radius of hemispherical ball = r
The rod is in equilibrium under the action of the following three forces as shown in Fig. 5.22.

1. Weight of the rod (W) acting vertically downwords through the mid-point G of the
   rod AB


2. Reaction at A acting in the direction AO


3. Reaction at C acting at the right angle to AB

Fig. 5.22.

From the geometry of the figure we know that

AD =2r

AC =AD cos θ = 2r cos θ

CD =AD sin θ = 2r sin θ

AG =GB = 1.5r

GC =AC – AG = 2r cos θ – 1.5r

From the geometry of the figure we also find that

∠GDC = θ

∴

2 cos – 1.5 (2 cos – 1.5)

tan tan

2sin 2sin

GC r r r

GDC

CD r r

θθ

θ= ∠ = = =

θθ

or

sin 2 cos – 1.5

cos 2 sin

θθ

=

θθ

2 sin^2 θ = 2 cos^2 θ – 1.5 cos θ

2 (cos^2 θ – sin^2 θ) = 1.5 cos θ

2 (2 cos^2 θ – 1) = 1.5 cos θ

4 cos^2 θ – 1.5 cos θ – 2 = 0