PROPERTIES OF MATTER BYU 17

Step 1: Establishing Force Equations

Let $W$ be the weight of the empty capsule and $B$ be the buoyant force (which equals the weight of the displaced liquid). Since the capsule volume is constant, $B$ is constant. Let the drag constant be $C$.

Case 1: Empty Capsule (Moves Up)
Upward Buoyancy opposes Weight and Drag.

$$ B – W = C v_1 \quad \text{…(i)} $$

Case 2: Completely Filled with Liquid (Moves Down)
If filled with the same liquid as outside, the weight of the liquid inside equals $B$. Total weight is $W + B$. Downward Weight opposes Buoyancy and Drag.

$$ (W + B) – B = C v_2 $$ $$ W = C v_2 \quad \text{…(ii)} $$

Substitute (ii) into (i):

$$ B – C v_2 = C v_1 \implies B = C(v_1 + v_2) \quad \text{…(iii)} $$

Part (a): $\eta$ Fraction Filled

The capsule is filled with fraction $\eta$ of the liquid. The weight of the added liquid is $\eta B$.
Total downward weight: $W_{total} = W + \eta B$.
Let the velocity be $v$ (assuming downward).

$$ F_{net} = W_{total} – \text{Buoyancy} = \text{Drag} $$ $$ (W + \eta B) – B = C v $$

Substitute $W$ and $B$ from equations (ii) and (iii):

$$ C v_2 + \eta C (v_1 + v_2) – C (v_1 + v_2) = C v $$

Divide by $C$:

$$ v_2 + \eta(v_1 + v_2) – (v_1 + v_2) = v $$ $$ v_2 + \eta(v_1 + v_2) – v_1 – v_2 = v $$

Note: If the result is negative, the capsule moves upwards.

Part (b): Filled with Liquid of Density $k$ times the Tube Liquid

If the density of the inner liquid is $k$ times the outer liquid, the weight of the filled portion is multiplied by $k$.
Weight of added liquid = $\eta (k B)$.

New Force Balance (assuming downward velocity $v’$):

$$ (W + \eta k B) – B = C v’ $$

Substituting expressions for $W$ and $B$:

$$ C v_2 + \eta k C(v_1 + v_2) – C(v_1 + v_2) = C v’ $$

Dividing by $C$ and simplifying:

$$ v_2 + \eta k (v_1 + v_2) – v_1 – v_2 = v’ $$