Answer to Question #130099 in Mechanics | Relativity for Ariyo Emmanuel

Question

A crane is used to lower weights into
the sea (density 1025 kg/m3
) for an
underwater construction project (Fig.
below). Determine the tension in the
rope of the crane due to a rectangular
0.4-m x 0.4-m x 3-m concrete block
(density 2300 kg/m3
) when it is (a)
suspended in the air and (b) completely
immersed in water

Answer 1

(a). Consider the free-body diagram of the concrete block. The forces acting on the concrete block in air are its weight and the upward pull action (tension) by the rope. These two forces must balance each other, and thus the tension in the rope must be equal to the weight of the block:

$F_T = W = mg = \rho Vg = 2300 \cdot 0.4 \cdot 0.4 \cdot 3 \cdot 9.81 = 10 \:830.24 \; N$ $\approx 10.8 \; kN$ .

(b). When the block is immersed in water, there is the additional force of buoyancy acting upward.

$F_B =\rho_{water} gV_{cube} = 1025 \cdot 9.81 \cdot 0.48 = 4826.5 \; N \approx 4.8 \; kN$.

The force balance in this case gives

$F_T = W - F_B= 10.8-4.8 = 6 \; kN$.

Answer: $6 \; kN.$