Variable Mass Systems: Rocket Propulsion and Falling Chains

1. The Correct Starting Point: Momentum

When mass is changing, the familiar form of Newton's second law, $\vec{F}=m\vec{a}$, becomes ambiguous. The more fundamental and universally applicable statement of the law is that the net external force on a system equals the rate of change of the system's total momentum:

The key challenge in variable mass problems is correctly defining the "system" and accounting for the momentum of every part, including the mass that is being ejected or accreted.

Let's use the product rule on $\vec{P}=m\vec{v}$:

It's tempting to set this equal to ${\vec{F}}_{\text{ext}}$, but this is often incorrect. This equation only describes the main body of mass $m$; it fails to account for the momentum carried away by the ejected mass (like rocket exhaust) or the momentum of mass about to be added. The proper way is to consider the total momentum of the main body and the mass elements it interacts with over a small time interval $dt$.

The following resource provides an excellent, rigorous derivation of the general equation of motion for a variable mass system.

The key result from this derivation is the general equation of motion for a variable mass system:

Where:

• $m$ and $\vec{v}$ are the instantaneous mass and velocity of the main body.
• ${\vec{F}}_{\text{ext}}$ is the net external force (like gravity or drag).
• ${\vec{v}}_0$ is the velocity of the mass element $dm$ before it is captured (for mass gain) or after it is expelled (for mass loss), as seen from a stationary reference frame.
• $\frac{dm}{dt}$ is the rate of change of the main body's mass. It's positive for mass gain and negative for mass loss.

The term $({\vec{v}}_0-\vec{v})$ is the velocity of the mass element $dm$ relative to the main body. Let's call it ${\vec{v}}_{\text{rel}}$. The equation can then be written more suggestively as:

The term ${\vec{v}}_{\text{rel}}\frac{dm}{dt}$ is often called the thrust force. It's not a new fundamental force, but rather a term that arises from correctly accounting for the momentum exchange in a variable mass system.

2. Application 1: Rocket Propulsion

Rocket propulsion is the classic example of a variable mass system where mass is continuously ejected. Let's analyze this using the principles we've just established.

The following video provides an excellent, step-by-step derivation for rocket motion, first by considering momentum conservation and then by offering a powerful alternative method that uses the concept of thrust directly.

As you saw, for a rocket moving vertically upward against gravity, the equation of motion is:

Notice that we use $-\frac{dm}{dt}$ because $\frac{dm}{dt}$ itself is negative (mass is decreasing), and the rate of fuel consumption is typically given as a positive value. Rearranging for integration gives:

Integrating this from an initial state ($t=0,v=v_i,m=m_i$) to a final state ($t,v_f,m_f$) yields the celebrated Tsiolkovsky Rocket Equation (with gravity):

This equation is the cornerstone of rocket science and a staple in advanced mechanics problems. Let's see how to apply it in a typical problem.

3. Application 2: A Falling Chain

Another classic variable mass problem, often found in Irodov, involves a chain falling onto a surface. Let's analyze the force exerted on the floor when a chain, initially held vertically, is released.

Suppose a chain of mass $M$ and length $L$ is dropped. What is the force on the floor at the instant a length $x$ of the chain has fallen?

It's tempting to say the force is just the weight of the portion of chain lying on the floor, which is $W_x=(\frac{M}{L})xg$. However, this misses a crucial component. The links of the chain are continuously crashing into the floor and coming to rest. This requires the floor to exert an upward impulsive force to stop them, and by Newton's third law, the chain exerts an equal and opposite force on the floor. This is a thrust force due to mass accretion by the stationary pile on the floor.

The video below gives a clear and concise derivation of the total force.

Let's summarize the logic:

1. Weight of the fallen part: $F_{\text{weight}}=(\text{mass per unit length})\times (\text{length on floor})\times g=\lambda xg$.
2. Thrust Force: The rate at which mass hits the floor is $\frac{dm}{dt}=\lambda \frac{dx}{dt}=\lambda v$. The change in momentum for each element $dm$ is from $v$ to 0. The force required is $F_{\text{thrust}}=\frac{dp}{dt}=v\frac{dm}{dt}=v(\lambda v)=\lambda v^2$.
3. Velocity: The chain is in free fall, so after falling a distance $x$, its velocity is $v^2=2gx$.
4. Thrust Force Calculation: Substituting $v^2$, we get $F_{\text{thrust}}=\lambda (2gx)=2\lambda gx$.
5. Total Force: The total force on the floor is the sum of these two forces. $$F_{\text{total}}=F_{\text{weight}}+F_{\text{thrust}}=\lambda xg+2\lambda xg=3\lambda xg$$

The total force is three times the weight of the portion of the chain already on the floor. This is a powerful and non-obvious result that stems directly from a correct analysis of a variable mass system.

Conclusion

Today we have ventured beyond the simple $F=ma$ to tackle the more complex and realistic world of variable mass systems. By starting from the fundamental principle of momentum conservation, we derived a general equation of motion and applied it to solve two cornerstone problems in advanced mechanics.

Here are the key takeaways:

• The fundamental equation for variable mass systems is $m\vec{a}={\vec{F}}_{\text{ext}}+{\vec{v}}_{\text{rel}}\frac{dm}{dt}$.
• The term ${\vec{v}}_{\text{rel}}\frac{dm}{dt}$ represents the thrust force, which arises from the momentum exchange with the ejected or accreted mass.
• For rocket propulsion, this leads to the Tsiolkovsky rocket equation, which relates the change in velocity to the exhaust velocity and the mass ratio.
• For a falling chain, the total force on the surface is the sum of the weight of the accumulated mass and a thrust force due to the continuous impact of new mass.

Mastering this topic is crucial for tackling many advanced problems. You now have the foundational tools to analyze any situation where mass is not constant.

In our next lesson, we will pivot back to rotational dynamics. We'll leverage the multivariable integration skills you developed in the first module to calculate the moment of inertia for complex continuous bodies, such as cones and hemispheres.