Question 9 (i):
A sinker is found to weigh 56.7 gf in water. When the sinker is tied to a cork of weight 6 gf, the combination is found to weigh 40.5 gf in water. Calculate the relative density of cork.
Solution to Question 9 (i)
Given:
- Weight of the sinker in water,
\( W_{s,w} = 56.7 \text{ gf} \) - Weight of the sinker + cork in water,
\( W_{\text{comb},w} = 40.5 \text{ gf} \) - Weight of the cork in air,
\( W_c = 6 \text{ gf} \)
Step 1: Determine the net upward force due to the cork
When the cork is tied to the sinker, the scale reading
drops from \(56.7 \) gf to
\(40.5 \) gf in water. The decrease
\( 56.7 – 40.5 = 16.2 \text{ gf} \)
is the net upward force exerted by the cork on the sinker.
Let \( F_B \) be the buoyant force on the cork
(weight of water displaced), and \( W_c = 6\text{ gf} \)
be the cork’s own weight. Then,
\( F_B – W_c = 16.2 \text{ gf} \).
Step 2: Find the buoyant force and the volume of the cork
Rearrange to solve for the buoyant force,
\( F_B = 16.2 \text{ gf} + W_c = 16.2 + 6 = 22.2 \text{ gf}. \)
By Archimedes’ Principle, this buoyant force is equal to the weight
of water displaced by the cork. Since the density of water is
\(1 \text{ g/cm}^3\), the
weight (in gf) of water displaced numerically equals
its volume (in cm3):
\[
\text{Volume of water displaced}
= F_B
= 22.2 \,\text{cm}^3.
\]
Since the cork is fully submerged (tied to the sinker), the volume
of the cork
\(V_{\text{cork}}\)
is
\(22.2 \text{ cm}^3\).
Step 3: Calculate the relative density of the cork
The (true) density of the cork is its mass divided by its volume.
The mass of the cork
\(m_c\)
is
\(6 \text{ g}\)
(because 1 gf ≈ 1 g under standard gravity).
\[
\rho_{\text{cork}}
= \frac{m_c}{V_{\text{cork}}}
= \frac{6 \text{ g}}{22.2 \text{ cm}^3}
\approx 0.27 \text{ g/cm}^3.
\]
Relative density (RD) is the ratio of the density of the cork to
the density of water (1 g/cm3):
\[
\text{RD}
= \frac{\rho_{\text{cork}}}{\rho_{\text{water}}}
= \frac{0.27}{1}
= 0.27.
\]
Final Answer
The relative density of the cork is
\( \boxed{0.27} \).