---
title: "Force by vane on steady stream"
subject: "Applied Mechanics (Dynamics)"
chapter: 3
key: "semester_3/applied_mechanics_dynamics:steady-stream-force-vane-deduce"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/steady-stream-force-vane-deduce
site: BCT Engineering
---

# Force by vane on steady stream

Applied Mechanics (Dynamics) · Chapter 3: System of particles

- Asked in: 2072 Chaitra, 4 marks
- Video: [Dynamics - steady stream of particles |finding force exerted by the blade on the stream](https://www.youtube.com/watch?v=K70GAfmaUC4) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/steady-stream-force-vane-deduce
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-3

## Question

Deduce an expression which shows the relation for the force exerted by the vane on the stream while you are dealing with the steady stream of particles.

## Short answer

For steady flow, resultant force on stream equals mass flow rate times velocity change: F = ṁ(v_exit – v_entry) = ρQ(v_B – v_A). Components F_x = ṁ(v_Bx – v_Ax), F_y = ṁ(v_By – v_Ay). For moving vanes use relative velocities and corresponding mass flow.

## Answer

A **steady stream** is a continuous flow in which the same mass of particles passes through any section per second, so the momentum of the particles inside the control region (between the entry section A and the exit section B) does not change with time.

```text
     entry A          vane           exit B
  v_A -->  ======== curved ========  --> v_B
           (mass flow m_dot in = out)
```

#### Derivation
Consider the particles in the portion of the stream between A and B at time $t$, plus the mass $\Delta m$ that enters at A during $\Delta t$.

- Momentum at $t$: $\vec H_{AB}+\Delta m\,\vec v_A$, where $\vec H_{AB}$ is the momentum of the particles between A and B.
- Momentum at $t+\Delta t$: $\vec H_{AB}+\Delta m\,\vec v_B$ (the mass $\Delta m$ has left at B; the contents between A and B are unchanged in steady flow).

The impulse-momentum principle: momentum at $t$ + impulse of the external forces = momentum at $t+\Delta t$:

$$
\vec H_{AB}+\Delta m\,\vec v_A+\vec F\,\Delta t=\vec H_{AB}+\Delta m\,\vec v_B
$$

$$
\vec F\,\Delta t=\Delta m(\vec v_B-\vec v_A)
$$

Dividing by $\Delta t$ and using $\dot m=\dfrac{dm}{dt}=\rho Av$:

$$
\boxed{\vec F=\dot m(\vec v_B-\vec v_A)=\rho Q(\vec v_B-\vec v_A)}
$$

where $\vec F$ is the resultant force exerted by the vane on the stream (and $-\vec F$ is the force of the stream on the vane) and $Q=Av$ is the discharge.

- Components: $F_x=\dot m(v_{Bx}-v_{Ax})$, $F_y=\dot m(v_{By}-v_{Ay})$.
- For a vane moving with speed $V$, the same result holds with $\vec v_A,\vec v_B$ taken relative to the vane and $\dot m=\rho A(v_A-V)$ for a single moving vane.

## Check yourself

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

1. What defines a steady stream?
   A. Constant speed of every particle
   B. Same mass passes each section per second and momentum between the sections does not change
   C. Zero external force
   D. Constant pressure only
   Answer: B. In steady flow, the momentum within the control region is unchanged with time, so only the entering and leaving mass matter. (Key point: Steady stream)

2. What causes the change of momentum over delta t?
   A. Delta m leaves at B with v_B while an equal delta m entered at A with v_A
   B. The contents between A and B change
   C. The vane's mass
   D. Gravity of the contents
   Answer: A. Net change = delta m (v_B - v_A); the content between A and B is unchanged. (Key point: Momentum change)

3. Which expression results?
   A. F = m dot (v_A - v_B)
   B. F = m dot v_B
   C. F = rho Q v_A
   D. F = m dot (v_B - v_A) = rho Q (v_B - v_A)
   Answer: D. Dividing F delta t = delta m (v_B - v_A) by delta t gives F = m dot (v_B - v_A). (Key point: Final expression)

4. What is the force of the stream on the vane?
   A. Equal to F and in the same direction
   B. -F, equal and opposite to the force on the stream
   C. Zero
   D. Half of F
   Answer: B. By Newton's third law the stream pushes on the vane with -F. (Key point: Force on vane)

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