What is Variable mass: Definition and 58 Discussions
In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system. Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as
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{\displaystyle \mathbf {F} _{\mathrm {ext} }+\mathbf {v} _{\mathrm {rel} }{\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}
where Fext is the net external force on the body, vrel is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body, and v is the velocity of the body. In astrodynamics, which deals with the mechanics of rockets, the term vrel is often called the effective exhaust velocity and denoted ve.
How to handle a variable mass system with Lagrangian mechanics? As far as I understand Newtonian mechanics fails, because the object is not constant anymore, it is updated every moment to a new object with different physical properties. I don't immediately see how Lagrangian mechanics can do better.
So i got some equations but i think i am missing something, my main doubt is what is the relation between dx / dt and v(o) [ here] . Workings in attachment
I chose to set the upwards direction to be positive and dM/dt = R = 190 kg/s, so I can solve the problem in variable form and plug in. With the only external force being gravity, this gives
M(t) * dv/dt = -M(t) * g + v_rel * R
where M(t) is the remaining mass of the rocket. Rearranging this...
The equations of variable mass systems are usually deduced from some very informal argument. It is so at least for the books I know.
So I tried to construct a formal proof based on the continuous media equations.
Criticism, remarks etc are welcomed.
Let ##D\subset \mathbb{R}^q## be an open...
This question stems from one of the recent homework threads. I'm familiar with the derivation given here regarding mass accretion and ejection, where the general idea is to define a system around body and all of the incoming/leaving mass so that we can once again apply NII to the whole thing.
I...
Hello,
I've got to rationally analice the form of the solutions for the equations of motion of a simple pendulum with a varying mass hanging from its thread of length ##l## (being this length constant).
I approached this with lagrangian mechanics, asumming the positive ##y## direction is...
The first way to solve this is to just say, by conservation of momentum, that M_{0}v_{0}=(M_{0}+Apx)\frac{v_0}{2} where Apx is the mass of dust the rocket comes into contact with in a distance x.
For the second method, by considering the change of momentum of the dust in 1 second, we know the...
In the book by Tipler & Mosca, the section on F=ma for variable mass derives the following equation:
##\mathbf{F}_{ext}+\frac{dM}{dt} \mathbf{v}_{rel}=M\frac{d\mathbf{v}}{dt}##
where ##\mathbf{F}_ext## is the external force on the system as a whole (ie not just the variable mass sub-system...
I was just doing some review on my physics lecture and I stumble on the idea of what if there was an object hanging and the cord mass is also included in the weight and it's displaced upward without having velocity nor time hypothetically and the cord change mass. I tried solving it by W = fΔx...
Homework Statement
A water tank with a total mass of m0 is moving on a horizontal road with a coefficient of friction equals to μ.
At t=0 water starts to come out of the tank with a velocity equal to u0 in relation to the tank. Each second the mass of water that comes out is λ.
Find the...
Homework Statement
A cart with no motor moves on a plane. At t=0 it has a mass equal to m0 and some velocity. Each second sand with a mass of α comes out of the cart with a velocity of 0 in relation to the plane. What is the equation of motion of the cart?
Answer: dV/dt=αv(t)/(m0-αt)
Homework...
Homework Statement
We have an Atwood machine like the picture below. one side (left) is a bucket full of water which has a hole on the bottom and the water is flowing with rate ##dm/dt = \alpha = const##. The initial mass of bucket with the water is ##m_0##. On the other side (right) we have a...
Homework Statement
A spigot pours beans onto a scale platform.
At a time t = 0.0 sec, the spigot is opened and beans begins to pour out (with initial velocity 0) at a rate of 1.00 kg/sec onto the platform from a height of 10.0 m above.
(a) At t = 10.0 sec, what is the weight of beans on the...
In classical mechanics, there's some debate about whether to define force as ##\mathbf{f} = m\mathbf{a}## or as ##\mathbf{f} = \dot {\mathbf{p}} = d (m\mathbf{v})/dt##. The former is much more popular I think, and it has the virtue of always having a Galilean-invariant magnitude ##f##, whereas...
Homework Statement
A locomotive is dragging empty freight cars, while coal is being dropped into them. It’s falling down into those freight cars with an efficiency (μ). Overall mass of the whole empty train is M.
a) Calculate v(t) (velocity with respect to time), assuming that the force of...
Homework Statement
Rockets are propelled by the momentum of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that for a Rocket starting...
Homework Statement
A pickup truck is driving at night through the rain. He drives onto a frozen lake. He immediately turns off his engine to save fuel (it wouldn't help anyway on frictionless ice) and let's the truck coast (move under no power). Unfortunately, the rain is accumulating in the...
Hey guys, I just wanted to check if my method for solving this problem is correct.
1. Homework Statement
Consider an annulus with inner radius R1 and outer radius R2. The mass density of the annulus is given by σ(r)=C/r, where C is a constant. Calculate the total mass of the annulus.
Homework...
Homework Statement
An armoured car with a mass of 5 tonnes is located on a smooth plane which is inclined at an angle of tan-1 (5/12) to the horizontal as shown in Figure Q3. A missile of mass 15kg is fired horizontally from this armoured car at 650m/s. Determine the velocity with which the...
Homework Statement
Suppose a rain drop with mass ##m_0\neq 0## is falling due to gravity with initial velocity ##v_0##, assume ##\frac{dm}{dt}=k=##constant. Solve the differential equation and determine the velocity as ##t\to\infty##
Homework Equations
##F=\frac{dp}{dt}=\dot{m}v+m\dot{v}##
The...
Is it really possible for a system to decrease its velocity with no forces acting on it, just because the mass in it is "varying"?
Consider for example a freight car and a hooper from which sand is released into the car. The freight car will decrease its initial velocity if there is no force...
I do not get why systems such as the rocket in space are defined as "variable mass" since the mass of the system is not varying.
The equation used for such systems $$\sum F^{(E)}=\frac{d\vec{P}}{dt} \tag{1}$$ (sum of external forces on the system equals the change in momentum) holds true only...
Homework Statement
A rocket with an initial mass of 60,000kg ignites its engines and burns fuel at a rate of 300 kg/s with an exhaust velocity of 2220 m/s. How long after the engines start does the rocket lift off the ground?
Homework Equations
From Newton's second law
F = Ma this equation can...
I am thinking about something that is getting me quite confused. To illustrate what is confusing me let's have a cart moving in the horizontal direction on a track without friction, and a motor which adjusts to keep the kart at a constant speed V in the positive direction. Now, it's snowing so...
Homework Statement
I am trying to derive the formula for time varying thrust given that I know the initial mass/volume of water in a water rocket. Knowns will include initial pressure, initial volume of water, and nozzle cross sectional area.
Homework Equations
Bernoullis equation: (p/ρg) +...
Hey guys,
I'm reading my modern physics book over break and I remember hearing that mass changes as you approach the speed of light. But is it really the mass that is changing or just the amount of momentum a certain mass can have. So is mass really varying or is it the energy capacity of mass...
Hello everyone.
I've been struggling with how to deal or solve questions that include variable masses. Considering the method I usually try to solve with, it either gets me lost or just makes things complicated and doesn't work often to me. :P
I was wondering if you guys could help me out...
A cart has a load of 100 kg sand at start and it is pulled by a steady resultant force of 100N starting from rest. A hole in the cart causes steady leakage of 1kg/s. Find velocity v of the cart after 50s of motion.
What I know about this is that, we have to use calculus.
dF =(dM /dT) *(dV/dT)...
A cart has a load of 100 kg sand at start and it is pulled by a steady resultant force of 100N starting from rest. A hole in the cart causes steady leakage of 1kg/s. Find velocity v of the cart after 50s of motion.
I'm a little confused about these.
Sometimes I have seen solutions where F=d(mv)/dt=mdv/dt+vdm/dt is used and solved as a differential equation. An example is this:
A water drap falls through a cloud. It has initial mass m which increases at a constant rate km as it falls. Show that it's...
Homework Statement
At time t=0 a dust particle of mass m_0 starts to fall from rest through a cloud. Its mass grows exponentially with the distance fallen, so that after falling through a distance x its mass is m_0exp[αx] where α is constant. Show that at time t the velocity of the particle...
I'm having some issues with the rocket equation. I'm deriving the velocity as a function of time for a descending rocket (so the rocket is accelerating upwards in order to slow the descent). The result I should obtain is
v=v0+gt+uln(mf/mi)
where mf is the final mass, mi is the initial...
Homework Statement
Water vapor condense in a raindrop with rate μ units of mass per time.
The raindrop starts with 0 velocity with initial mass $$ M_0 $$ and falls horizontally. Find the space it traverses as a function of time (g is given).Homework Equations
∂m/ ∂t= μ
The Attempt at a...
How do we know that the work-energy theorem holds for variable mass systems? Or rather, since I'm sure that we can at least know it to be true experimentally, what is the mathematical basis for the work-energy theorem? I know for fixed mass systems, a rather simple derivation comes from...
Homework Statement
A flatcar of mass ##m_0## starts moving to the right due to a constant horizontal force F. Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to ##\mu## kg/s. Find the time dependence of the velocity and acceleration of flatcar...
Homework Statement
Figure 5-47 shows Atwood's machine, in which two containers are connected by a cord (of negligible mass) passing over a frictionless pulley (also of negligible mass). At time t = 0, container 1 has mass 1.30 kg and container 2 has mass 2.80 kg, but container 1 is losing...
If a railroad car traveling down a straight, frictionless track encounters vertical rain that fills it with additional mass, will the velocity decrease in order for the momentum to be conserved? Will the opposite happen if the water is let out from a hole in the bottom?
I cannot see any...
Homework Statement
An open-topped railway wagon of mass M is rolling freely along straight level frictionless track at a constant velocity v0. At time t=0 the wagon enters a heavy rain shower and starts to collect rainwater. The rain falls vertically. As a consequence the wagons mass increases...
Homework Statement
A 2kg bucket containing 10kg of water is hanging from a vertical ideal spring of force constant 125N/m and oscillating up and down with amplitude 3cm. Suddenly the bucket springs a leak in the bottom such that water drops out of the bucket at a steady rate of 2g/s...
Homework Statement
Water with mass density ρ are being shoot from a jet with cross sectional area A and velocity v0 at a man with mass m0.
1. Find the force acting on a man at rest (assume all the water are absorbed in the clothes)
2. Find the force acting on a man escaping at velocity v<v0...
So I was sitting on the train last weekend, reading through my physics book on mechanical work and its relation to kinetic energy. One example would be that a box on a frictionless table being pushed and they would conclude that W = ΔK = ½mΔv2.
Looking at this equation got me thinking...
Homework Statement
I've pretty much solved it, but I'm unsure of my final integration
A uniform chain of length L and density /rho(kg/m) is initially stationary on a horizontal, frictionless table, with part of the chain (length yo) hanging over the edge. How much time passes before the...
Homework Statement
A meter stick has constant thickness and width , but the material that the stick is constructed from is very strange ... it has a variable mass density that is given by, ρ(x) = 0.800(1 + 0.00250x) grams/cm3 where x is measured in cm. Find the center of mass of the meter...
Thread split from this thread.
Not true.
The spray can is generating a constant force, not a constant power. As a result, the work done by the spray can depends on the distance it travels, and the power generated depends on the rate at which it is traveling. The farther the spray can travels...
The following question, although probably sounding inappropriate for this forum, is actually very important to me. The easy answer to the question is, “It can’t be done.” But a great deal of learning and increased understanding could be gained if the answer were, “It could be done if...”...
Homework Statement
a heap of chain is lying on a table with a hole. A small part of chain is released through the hole.calculate the velocity as a function of the length of the chain hanging vertically.
[Homework Equations
The Attempt at a Solution
if I take a mass element dm...
I'm curious - what would happen if a satellite was in orbit around an object which suddenly lost a large piece of its mass, or gained a large amount of mass? Of course this seems extremely unlikely to occur in nature, but I suppose similar reactions could be produced by using engines to counter...