# Homework 5 - Aidan Sharpe

## 1
If the specific weight, $\gamma$, of a substance is 8.2[kN/m$^3$], what is its mass density, $\rho$, in kg/m$^3$?

On Earth, 9810[kg] weighs 1[kN], so an object that weighs 8.2[kN] will have a mass of 80442[kg]. Therefore, $\rho = 80442$[kg/m$^3$].

## 2
A fluid flowing between two parallel plates has a viscosity, $\mu = 0.62$[Ns/m$^2$], and density, $\rho = 1250$[kg/m^3]. Calculate the intensity of shear stress, $\tau$, in pascals at $y = 3$[cm], assuming a straight-line viscosity distribution, given that the top plate has a velocity of 100[cm/s] and the fluid is 6[cm] thick.

Velocity at $y = 3$[cm]:
$$\tau = \mu{du \over dy} = 0.62\left(0.03 \times {1 \over 0.06}\right) = 0.31\text{[Pa]}$$

## 3
A cube with side length 10[cm] is placed into two different liquids. In the first liquid, the top of the cube is $h_1 = 1$[cm] above the surface. In the second liquid, the top of the cube is $h_2$ above the surface. If the densities of the liquids are known to be $\rho_1 = 1000$[kg/m$^3$], and $\rho_2 = 1300$[kg/m$^3$] respectively, find $h_2$.

Displaced mass for liquid 1, same as mass of cube:
$$0.1 \times 0.1 \times (0.1 - 0.01) \times 1000 = 0.9\text{[kg]}$$

Displaced mass for liquid 2:
$$0.1 \times 0.1 \times (0.1 - h_2) \times 1300 = 0.9$$
$$\therefore h_2 = 0.031\text{[m]} = 3.1\text{[cm]}$$