A block of wood 6 ft by 8 ft by 10 ft floats on oil of sp gr 0.751. A clockwise couple holds the block in the position shown in Fig. 4-9. Determine (a) the buoyant force acting on the block and its position, (b) the magnitude of the couple acting on the block, and (c) the location of the metacenter

Question

Accepted Answer

(a) weight of block = weight of triangular prism of oil (or the buoyant force)

[latex]W=F^{′}_{B}= (0.751 imes 62.4)\left(\frac{1}{2} imes 8 imes 4.618 imes 10 ight)=8656 [/latex] lb

Then F′_B = 8656 Ib acting upward through the center of gravity O′ of the displaced oil. The center of gravity lies 5.333 ft from A and 1.540 ft from D, as shown in Fig. 4-9.

AC = AR + RC = AR + LO′ = 5.333 cos30° + 1.540 sin30° = 5.389 ft

The buoyant force of 8656 Ib acts upward through the center of gravity of the displaced oil, which is 5.39 ft to the right of A.

(b) One method of obtaining the magnitude of the righting couple (which must equal the magnitude of the external couple for equilibrium) is to find the eccentricity e. This dimension is the distance between the two parallel, equal forces W and F′_B that form the righting couple.

e = FC = AC - AF = 5.389 - AF = 5.389 - 4.963 - 0.426 ft

since AF = AR + RF = 4.618 + (0.691)(sin30°) = 4.963 ft

The couple We or F′_B e = 8656 x 0.426 = 3687 ft-lb. Thus the moment or couple to hold the block in the position shown is 3687 ft-lb clockwise.

(c) The point of intersection of the buoyant force and the axis of symmetry SS is called the metacenter (point mc in the figure). If the metacenter is located above the center of gravity of a floating object, the weight of the object and the buoyant force form a righting moment in tilted positions.

The metacentric distance = [latex]\frac{RC}{\sin 30^\circ }-0.691=\frac{0.770}{0.5000}-0.691 =0.849 [/latex] ft.

In naval architecture, an extreme angle of some 10° is taken as the limit of heel for which the metacentric distance can be considered constant.

Question 4.12: A block of wood 6 ft by 8 ft by 10 ft floats on oil of sp gr...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions