- #1
brotherbobby
- 699
- 163
- Homework Statement
- Rank in order, from largest to smallest, the ##\mathbf{densities}## of blocks a, b and c shown in figure below. Explain your reasoning.
- Relevant Equations
- Law of floatation : "The weight (or mass) of a floating body is equal to the weight (or mass) of the liquid displaced : ##w_B = \Delta w_L##
The dimensions of the bodies given in the problem are visual. Clearly bodies a and b are cubical whereas c is not. One side of b is twice the side of a. Both a and b are submerged to the same depth but what is the depth of submersion of c? Arranging bodies (by copying and pasting a on b and c, as can be done using imaging programs), I paste the following diagram that show the depths and sides of the bodies shown. For simplicity, let us ignore the third dimension of the bodies or assume them to be equal, equivalently.
Please note the case for c . Also we have ##y < x##.
Using ##w_{\text{body}} = \Delta w_L## and dropping the (constant) density of the liquid, we have for the weights of the three bodies :
##w_a = xy,\; w_b = 2xy,\; w_c = 2xy##.
As for the volumes of the three bodies,
##v_a = x^2,\; v_b = 4x^2,\; v_c = x(2x+y)##.
Dividing the first by the second and supressing the acceleration due to gravity ##g## from each, we get the densities of the three bodies,
##\rho_a = \frac{y}{x}, \; \rho_b = \frac{y}{2x}, \rho_c = \frac{2y}{2x+y}##.
Clearly, ##\boxed{\rho_b < \rho_a}##.
Using simple algebra and remembering that ##y<x## yields the solution :
##\large{\boxed{\mathbf{\rho_a > \rho_c > \rho_b}}}##Of course I'd like to know if I am right. Even if I am, is there an easier and a more conceptual way to do this?