Fluids and Newton’s Laws

Learning Goals

After this lesson, you should be able to:

• choose a fluid particle, fluid region, or immersed object as a system and identify external forces;
• explain how internal particle interactions and external forces produce a fluid's macroscopic behavior;
• construct free-body diagrams for submerged, suspended, and floating objects;
• derive buoyant force from the pressure difference across an immersed object;
• apply Archimedes' principle using displaced-fluid volume;
• predict whether an object initially rises, sinks, or remains at rest;
• calculate apparent weight, string tension, payload, and fraction submerged;
• distinguish Newton's third-law partners from forces that balance on one object; and
• design and evaluate buoyancy experiments.

1. Newton's Laws Still Govern Fluids

A fluid is made of particles, and each particle obeys Newton's laws. For any selected particle or collection of particles,

If the net external force is zero, the chosen system's center-of-mass velocity remains constant. If forces are unbalanced, its velocity changes. The challenge is selecting a useful system boundary and combining many particle interactions into macroscopic pressure forces.

2. Microscopic Interactions Become Macroscopic Forces

Neighboring fluid particles exchange forces during interactions. When the system contains a large region of fluid, forces between particles inside the boundary are internal and occur in Newton's third-law pairs. They cancel from the net force on the whole region.

Forces from container walls, surrounding fluid across the system boundary, gravity, and immersed objects are external. These external forces determine the acceleration of the selected fluid region.

4. Pressure Forces on an Immersed Object

Each small surface patch feels a force normal to that patch. Horizontal components on opposite sides cancel in a symmetric static-fluid situation. Because pressure increases with depth, the bottom experiences a larger upward force than the downward force on the top.

The vector sum of all fluid forces is the buoyant force. For an object in a fluid under ordinary gravity, that net force points upward.

5. Deriving Buoyant Force for a Cylinder

Consider a fully submerged vertical cylinder of cross-sectional area \(A\) and height \(H\). Its top and bottom pressures differ by

The horizontal pressure forces cancel, so the net vertical fluid force is

Since \(HA=V_{\mathrm{cylinder}}=V_{\mathrm{displaced}}\),

The reference pressure cancels. Atmospheric pressure can be large without adding a separate term to the buoyant-force result.

6. Archimedes' Principle

The cylinder result generalizes to an object of any shape:

The density is the density of the surrounding fluid. The volume is the volume of fluid pushed aside, not automatically the object's total volume.

7. Draw the Object's Free-Body Diagram First

With upward positive, a freely moving immersed object commonly has

If a string, support, or drag force is present, add it separately. Buoyancy and weight are not a Newton's third-law pair because they both act on the same object. Their comparison determines the object's net force.

8. Rise, Sink, or Neutral Buoyancy

“Floats” usually describes equilibrium at a surface; \(F_B>mg\) describes an initially rising, fully immersed object before it reaches that equilibrium.

9. Density Test for a Fully Submerged Object

For a completely submerged object, \(V_{\mathrm{displaced}}=V_{\mathrm{object}}\). Then

• If \(\rho_{\mathrm{obj}}<\rho_{\mathrm f}\), the initial net force is upward.
• If \(\rho_{\mathrm{obj}}=\rho_{\mathrm f}\), the object is neutrally buoyant.
• If \(\rho_{\mathrm{obj}}>\rho_{\mathrm f}\), the initial net force is downward.

Use average object density when the object is hollow or made of several materials.

10. Worked Example: Initial Acceleration

A sealed object of mass \(1.60\,\mathrm{kg}\) and volume \(2.00\times10^{-3}\,\mathrm{m^3}\) is fully submerged in water and released from rest. Ignore drag during the initial instant.

Answer: The initial acceleration is \(2.45\,\mathrm{m/s^2}\) upward. Later, drag may change the acceleration, so this is not necessarily the acceleration throughout the rise.

11. Buoyant Force and Depth

For a fully submerged rigid object in a constant-density fluid,

None of these factors changes when the object is moved deeper, so the buoyant force remains constant even though absolute pressure on every part of the object increases. Buoyancy depends on the pressure difference, and that difference across a fixed height remains \(\rho_{\mathrm f}gH\).

If fluid density changes with depth or the object compresses, this conclusion may no longer hold.

12. Partially Submerged Objects

For partial submersion, only the volume below the fluid surface displaces fluid:

As an object settles farther into the fluid, displaced volume and buoyant force increase. The object reaches floating equilibrium when the upward buoyant force equals its total weight.

13. Fraction Submerged for a Floating Object

For an object floating at rest with no other vertical forces,

A block with average density \(650\,\mathrm{kg/m^3}\) floating in water has \(65.0\%\) of its volume submerged. The result uses average density and assumes no additional vertical force.

14. Floating Payload and Displaced Mass

A floating system in equilibrium displaces a fluid mass equal to its total mass:

A small boat can displace at most \(0.018\,\mathrm{m^3}\) of fresh water before water reaches its rim. Its maximum supported total mass is

If the empty boat has mass \(5.0\,\mathrm{kg}\), the maximum payload in this ideal static model is \(13\,\mathrm{kg}\).

15. Apparent Weight and String Tension

A fully submerged object of mass \(3.00\,\mathrm{kg}\) and volume \(1.20\times10^{-3}\,\mathrm{m^3}\) hangs at rest from a scale in water.

The free-body equation is

Answer: The scale reads \(17.6\,\mathrm N\). “Apparent weight” here means the support force, not a change in the gravitational force.

16. Buoyancy Exists in Gases

Air is a fluid, so an object in air also displaces fluid and experiences buoyancy:

For compact dense objects, this is often small compared with weight. For a large low-density balloon, displaced-air weight can be large enough to exceed the combined weight of the gas, envelope, and payload, producing an upward net force.

17. Shape Changes the Force Distribution, Not Archimedes' Result

Changing an object's orientation changes which surface patches experience particular pressure forces. Nevertheless, for a completely submerged rigid object in a uniform fluid, the net buoyant-force magnitude remains

Shape and orientation can strongly affect drag during motion, but drag is a separate force from static buoyancy.

18. Newton's Third-Law Interaction Pairs

If the fluid exerts an upward buoyant force \(\vec F_{\mathrm{fluid\ on\ object}}\), the object exerts an equal-magnitude downward force on the fluid:

These forces act on different systems. By contrast, buoyancy and weight both act on the object and may balance, but they are not third-law partners.

A container scale can register the object's downward interaction transmitted through the fluid, provided the complete system and supports are analyzed.

19. What Makes a Fluid Accelerate?

Select a small region of fluid. External forces on it can include pressure forces from surrounding fluid, gravity, contact forces from walls, and forces from objects. If these do not sum to zero, the region accelerates:

In hydrostatic equilibrium, pressure forces and weight balance. A pressure imbalance or another unbalanced external force changes the fluid's velocity. The continuity and energy description of established flow is developed in Topic 8.4.

20. Experiment: Determine an Object's Density

Use a force sensor, an object that sinks, a string, and a liquid of known density.

1. Measure the object's weight in air, \(W\).
2. Submerge it completely without touching the container and record the steady tension \(T\).
3. Calculate buoyancy from \(F_B=W-T\).
4. Calculate displaced volume from \(V=F_B/(\rho_{\mathrm f}g)\).
5. Calculate object mass from \(m=W/g\) and density from \(\rho_{\mathrm{obj}}=m/V\).
6. Repeat at several depths to test the prediction that \(F_B\) is depth-independent.
7. Report sensor resolution and repeat-trial variation.

21. Experimental Errors and Model Limits

• Object touching the container adds an unmodeled normal force.
• Partial submersion makes displaced volume smaller than total object volume.
• Air bubbles increase displaced volume and the measured buoyant force.
• String buoyancy can affect a high-precision force measurement.
• Fluid motion adds drag, so take a steady reading for a static model.
• Object compression changes displaced volume with depth.
• Density gradients make buoyancy depend on location.
• Surface contact or surface tension can add forces not included in the simple two-force floating model.

22. A Reliable Buoyancy Workflow

1. Select the object or fluid region as the system.
2. Draw every external force acting on that system.
3. Identify the displaced-fluid volume; do not assume it equals total object volume.
4. Use the surrounding fluid's density in \(F_B=\rho_{\mathrm f}gV_{\mathrm{displaced}}\).
5. Apply \(\sum F_y=ma_y\) with a declared positive direction.
6. For floating equilibrium, set \(F_B=mg\) only if no other vertical forces act.
7. For a fully submerged object, compare average object density with fluid density.
8. Separate initial acceleration from later motion affected by drag.
9. Check units, limiting cases, and whether the static constant-density model is justified.

23. Common Traps and Key Takeaways

• Buoyancy is the net fluid force, not an extra force in addition to every pressure force.
• Buoyant force equals displaced-fluid weight, not object weight in every situation.
• Use fluid density, not object density, in \(F_B=\rho_{\mathrm f}gV_{\mathrm{displaced}}\).
• Use submerged volume for a partially immersed object.
• A sinking object still experiences buoyancy.
• For a fully submerged rigid object in uniform fluid, buoyancy is independent of depth.
• Floating equilibrium means \(F_B=mg\) when those are the only vertical forces.
• Apparent weight is a support-force reading, not altered gravity.
• Buoyancy and weight are not a third-law pair.
• Newton's laws apply to fluid particles and fluid regions.