E-9

Pre-lab exercises

1. The buoyant force experienced by the solid is W₂ - W₃
   V = W/r, (W₂ - W₃)/r = V_f
   V = V_f, hence W/r = (W₂ - W₃)/r_f. we have
   r = r_f[W/(W₂ - W₃)]
2. The density of the alloy is r = m/V, m = m₁ + m₂
   The volumes of the two metals are V₁ and V₂ respectively.
   V₁ = m₁/r₁, and V₂ = m₂/r₂.
   r = (m₁ + m₂)/(V₁ + V₂) = (m₁ + m₂)/(m₁/r₁ + m₂/r₂)
3. The specifiiic gravity is unitless

Calculation

7. Use modified equation (9-3): r_f = [r(W - W_app)]/W

Error Analysis

4. The expression can be derived as
r = r_fW/(W - W_app) = r_fW(W - W_app)^-1
s_r²/r² = s_W²/W² + s_(W-Wapp)²/(W - W_app)².
By using eq Z = ax + by + cz, we can estimate
s_{(W- Wapp)}² = s_W² + s_Wapp²,
since W and W_app are both measured with the same balance, their uncertainties, s_W are the same.
s_r²/r² = s_W²/W² +2s_W²/(W - W_app)²
= s_W²[(W - W_app)² + 2W²]/W²(W- W_app)² = s_W²(W² - 2WW_app + W_app² + 2W²)/W²(W - W_app)²
taking approximation 2WW_app » 2W² and r = r_fW/(W - W_app), we have
s_r² = r_f²(W² + W_app²)s_W²/(W - W_app)⁴

Answer to the question 1
In step 1, 2 we use r = m/V, where V = pa²L, but in step 3, 4 we use r = r_fW/(W - W_app), now you try to derive s_r in two different equations similar to the derivation that I did in Error analysis 4, then compare the results. You can tell which methode is more precise. This is a very interesting problem.