02 / CHAPTER 11
Variable-mass systems
See why rockets and leaks require momentum accounting, not a naive \(F=ma\).
ANCHOR EQUATION
Momentum balance
In symbols: d(mv)/dt = ΣFext
Variable-mass systems require momentum accounting rather than a blind F = ma substitution.
A leaking body needs momentum accounting
When mass crosses a system boundary, the momentum carried by the escaping or entering material contributes to the body’s momentum change. Treating the body as a fixed-mass particle silently loses that flux.
Differentiate m v
For a chosen body, p = mv, so dp/dt = m dv/dt + v dm/dt. External force equals the momentum change of the body plus the momentum exported through the boundary; the exact rocket equation also includes the relative exhaust velocity. For a rocket in empty space, thrust magnitude is approximately ṁ v_e, where ṁ is mass expelled per second and v_e is exhaust speed relative to the rocket.
If a pump ejects water at ṁ = 0.40 kg/s and changes its velocity by 12 m/s, the water momentum flux is 4.8 N. The equal and opposite reaction is the ideal force transmitted to the pump, before pressure and support effects.
Sign conventions carry the physics
The sign of ṁ depends on whether it enters or leaves the chosen system. A control volume is often safer than assigning a changing mass to a single-particle formula, especially when inlet and outlet velocities differ.
Work it: A 0.10 kg/s jet leaves a nozzle at 20 m/s while entering water is nearly at rest. Estimate the reaction force, then add a 5 N pressure-force contribution and state the total support load direction.