Answer to Question #275279 in Physics for marie

Question

A block of cherry wood that is 20.0 cm long, 10.0 cm wide, and 2.00 cm thick has a density of 800. kg/m3. What is the volume of a piece of iron that, if glued to the bottom of the block, makes the block float in water with its top just at the surface of the water? The density of iron is 7860 kg/m3, and the density of water is 1000. kg/m3.

Answer 1

Weight of the wood:

$$W_w=V_w\rho_wg=abc\rho_wg.$$

Weight of the iron:

$$W_i=V_i\rho_ig.$$

The buoyancy force:

$$F=(abc+V_i)\rho g,$$

must be equal to the combined weight of wood and iron:

$$(abc+V_i)\rho g=abc\rho_wg+V_i\rho_ig,\\\space\\ (abc+V_i)\rho =abc\rho_w+V_i\rho_i,\\\space\\ V_i=abc\frac{\rho-\rho_w}{\rho_i-\rho},\\\space\\ V_i=20·10·2·\frac{1000-800}{7800-1000} =11.8\text{ cm}^3.$$