---
title: "Hydrometer use and stability of floating bodies"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_3/fluid_mechanics:hydrometer-stability-floating-bodies"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/hydrometer-stability-floating-bodies
site: BCT Engineering
---

# Hydrometer use and stability of floating bodies

Fluid Mechanics · Chapter 3: Hydrostatics

- Asked in: 2074 Asoj, 6 marks
- Video: [Fluids, Buoyancy, and Archimedes' Principle](https://www.youtube.com/watch?v=16HDJNoXQII) — Professor Dave Explains
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/hydrometer-stability-floating-bodies
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-3

## Question

Explain the use of hydrometer and shortly explain the conditions of stability of floating bodies.

## Short answer

A hydrometer floats to measure liquid density; stem scale reads specific gravity. For floating bodies, stability requires metacentre M above centre of gravity G (GM > 0), giving a restoring moment W·GM sinθ.

## Answer

#### Hydrometer

A **hydrometer** is a floating instrument used to find the specific gravity (or density) of a liquid directly. It is a sealed glass tube with a weighted bulb at the bottom (so that it floats upright) and a graduated stem at the top.

**Principle:** it floats in equilibrium, so weight = buoyant force. Its weight $W$ is constant, so a denser liquid needs less displaced volume and the hydrometer floats higher; a lighter liquid makes it sink deeper.

$$
W = \rho g V_{sub}\quad\Rightarrow\quad \rho = \frac{W}{g V_{sub}}
$$

If the stem has area $a$ and sinks by an extra length $x$, then $V_{sub} = V_0 + a x$. The scale is calibrated in specific gravity, which is read at the liquid surface. The scale is not uniform: because the hydrometer floats higher in a denser liquid, the surface cuts the stem lower down, so the low values are at the top of the stem and the high values at the bottom, and the divisions get closer together towards the bottom (high-density end).

**Uses:** testing milk, acid, battery electrolyte, alcohol, sugar solutions and sea water.

```text
      |  <- stem with scale (reads S)
 ~~~~~|~~~~~ liquid surface
      |
    (   )  <- bulb
    (___)  <- lead shots (ballast)
```

#### Stability of floating bodies

A body floats in equilibrium when weight = buoyant force and the centre of gravity $G$ and the centre of buoyancy $B$ lie on the same vertical line. For stability, a small tilt must produce a restoring couple. The **metacentre M** is the point where the line of action of buoyancy through the new centre of buoyancy cuts the original vertical axis.

| Condition | Position | Result |
|---|---|---|
| Stable equilibrium | M above G ($GM > 0$) | Restoring couple, body returns |
| Unstable equilibrium | M below G ($GM < 0$) | Overturning couple, body capsizes |
| Neutral equilibrium | M coincides with G ($GM = 0$) | Body stays in the new position |

$$
GM = \frac{I}{V} - BG
$$

where $I$ is the second moment of the waterline area about the longitudinal axis, $V$ is the volume of liquid displaced and $BG$ is the distance between $B$ and $G$ ($G$ above $B$). The restoring moment is $W\cdot GM\sin\theta$. A body with a large metacentric height is more stable, but rolls quickly.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In a denser liquid, a hydrometer floats:
   A. Higher, since less volume is needed to displace its constant weight
   B. Lower
   C. At the same level
   D. Higher, because its weight increases
   Answer: A. W = ρ g V_sub with W constant, so higher ρ means lower V_sub. (Key point: Hydrometer principle)

2. What is the purpose of the weighted bulb with lead shot?
   A. To increase buoyancy
   B. To make the hydrometer float vertically
   C. To make the scale uniform
   D. To increase its volume
   Answer: B. The ballast lowers G below B for upright stability. (Key point: Ballast)

3. Which is correct for floating-body stability?
   A. M above G is stable; M below G is unstable; M at G is neutral
   B. M below G is stable
   C. M at G is unstable
   D. M above G is unstable
   Answer: A. The couple restores the body only if M lies above G. (Key point: Stability conditions)

4. How is GM computed?
   A. GM = V/I - BG
   B. GM = I/V + BG
   C. GM = I/V - BG, with I of the waterline area and V displaced volume
   D. GM = BG - I/V
   Answer: C. BM = I/V; GM = BM - BG when G is above B. A large GM gives stability but rapid rolling. (Key point: Metacentric height formula)

## Related in Fluid Mechanics

- [Stability of conical buoy with light](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aconical-buoy-stability) — Fluid Mechanics, chapter 3

## Related in other subjects

- [Buoyancy and stability of floating bodies](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Abuoyancy-metacentric-height-stability) — Fluid Mechanics, chapter 2