Unit 5 · Topic 06

Variable-mass systems

When mass crosses a system boundary, it carries momentum. The balance requires a declared boundary and velocities expressed in one coherent frame.

Unit 5Variable massOpen navigation

Concept 01

Mass entering or leaving

Essential The minimum you should retain

A variable-mass system exchanges matter and momentum with its surroundings.

UnderstandInterpret and connect

Using F=ma without a flow term can omit the dominant effect.

DeepenFormulation and conditions

First declare the boundary, mass flow rate, and inlet and outlet velocities.

ExploreConnections for further study

Different boundaries may describe the same case when all flows are counted.

Concept 02

Momentum rate

Essential The minimum you should retain

A steady one-dimensional flow obeys F=ṁ(v_out-v_in) F=\dot m(v_{out}-v_{in}) .

UnderstandInterpret and connect

The flow rate ṁ is positive and velocities carry signs in a declared frame.

DeepenFormulation and conditions

The product kg/s times m/s has units of newtons.

ExploreConnections for further study

In several dimensions, apply the vector balance to every inlet and outlet.

Worked example

Jet stopped by a plate

A jet carries 3.0 kg/s of water at +8.0 m/s and exits with zero horizontal velocity.

Given
  • ṁ=3.0 kg/s
  • v_in,x=+8.0 m/s
  • v_out,x=0
Target

Calculate the plate's force on the water.

  1. Model

    F_x=ṁ(v_out,x-v_in,x).

  2. Substitution

    F_x=(3.0)(0-8.0).

  3. Calculation

    F_x=-24 N.

  4. Third law

    Water exerts +24 N on the plate.

Conclusion

Force on the water is -24 N.

Mathematical relation

Steady momentum flow

F→=ṁ(vout−vin) \vec F=\dot m(\vec v_{out}-\vec v_{in})
Represents

Net force on a steady flow.

Physical interpretation

Changing transported momentum requires force.

DeepenVariables, conditions, and checks

Variables

ṁ
positive mass flow rate; usual unit: kg/s
v_in,v_out
velocities in one frame; usual unit: m/s

Conditions of application

  • Steady state and a declared boundary.

Dimensional check

kg/s·m/s=N.

A duct changes flow direction within a boundary.control volumeṁ v_inṁ v_outF

The difference between outlet and inlet momentum flows determines net force on the fluid.

Concept 03

Force on flow and support

Essential The minimum you should retain

The balance finds net force on material passing through the control volume.

UnderstandInterpret and connect

The fluid's force on a plate is opposite the plate's force on the fluid.

DeepenFormulation and conditions

Stopping or turning a jet requires changing its momentum per unit time.

ExploreConnections for further study

Separate free-body diagrams for fluid and support prevent reversing the answer.

Concept 04

Outlet velocity matters

Essential The minimum you should retain

Losing mass does not automatically accelerate or slow the remaining body.

UnderstandInterpret and connect

If mass leaves with the cart's instantaneous horizontal velocity, it carries no extra relative horizontal momentum.

DeepenFormulation and conditions

Acceleration depends on relative velocity across the boundary.

ExploreConnections for further study

This prepares propulsion without naively applying a constant-mass equation.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using F=ma on a variable-mass portion without momentum flow.

Account for mass crossing the boundary and its velocity.

Mixing velocities measured in different frames.

Express v_in and v_out coherently in a declared frame.