Momentum Definition and 1000 Threads

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity (also a vector quantity), then the object's momentum is





p

=
m

v

.


{\displaystyle \mathbf {p} =m\mathbf {v} .}
In SI units, momentum is measured in kilogram meters per second (kg⋅m/s).
Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry.
Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is generalized momentum, and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function. The momentum and position operators are related by the Heisenberg uncertainty principle.
In continuous systems such as electromagnetic fields, fluid dynamics and deformable bodies, a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.

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  1. S

    Momentum Conservation: How to Reconcile a Negative Value?

    Maybe a silly question but on the above question using the conservation of momentum: momentum before firing (0) = momentum after firing (55*35)+(M*2.5) If I re-range the above it's M = -(55*35)/2.5 = -770kg. I can I reconcile that minus sign (basically get rid of it)? Thanks
  2. M

    I Changes in angular momentum for ro-vibrational transitions

    Hello! If we have a transition between 2 ro-vibrational levels of the same electronic state of a diatomic molecule the selection rules require for the changes in the rotational quantum number J that ##\Delta J = \pm 1##. Why can't we have ##\Delta J = 0##? The photon carries one unit of angular...
  3. tanaygupta2000

    Expectation value of momentum operator

    I know that the eigenstates of momentum operator are given by exp(ikx) To construct a real-valued and normalized wavefunction out of these eigenstates, I have, psi(x) = [exp(ikx) + exp(-ikx)]/ sqrt(2) But my trouble is, how do I find the expectation value of momentum operator <p> using this...
  4. L

    I 1D wavepacket scattering simulation, momentum distribution formula

    Hello everybody at the forum I'm from Ukraine, I have Chemistry degree, and last year I began to self studying Quantum Mechanics. I'm reading this article: R. Garcia, A. Zozulya, and J. Stickney, “MATLAB codes for teaching quantum physics: Part 1,” [Online]. Available...
  5. L

    Elastic Collision and Momentum of Ice Skaters

    1. Hello, so the difficulty I am having with this problem is that is seems relatively straightforward. I have tried to solving it by assuming that this is a collision in which momentum is conserved. Therefore, I found the total momentum before the collision and used this to resolve it must be...
  6. agnimusayoti

    Invariance of Energy Momentum Relativistic

    I try to use relativistic energy equation: $$E'=\gamma mc^2$$ But, I use $$\gamma=\frac{1}{\sqrt{(1-(\frac{v'}{c})^2}}$$ then I use Lorentz velocity transformation. $$v'=\frac{v-u}{1-\frac{uv}{c^2}}$$ At the end, I end up with messy equation for E' but still have light speed c in the terms. How...
  7. yucheng

    What's the error in my solution (Freight car and hopper)

    Textbook solution: ##v## is the instantaneous velocity, $$P(t)=(M+b t) v$$ Then $$impulse = \Delta P = (M+b t) v = \int^{t}_{0} F dt'$$ Thus $$v=\frac{F t}{(M + bt)}$$ What I did instead was: Let ##M## be the instantaneous mass, and ##M_0## be the initial mass, then $$M=M_{0} + b t$$...
  8. yucheng

    Rocket propulsion equation: what's the error here?

    Equation for rocket motion: $$\frac{d \mathbf{P}}{d t} = M\frac{d \mathbf{v}}{d t} - \mathbf{u}\frac{d M}{d t}$$ But $$\mathbf{F}=\frac{d \mathbf{P}}{d t}=M\frac{d \mathbf{v}}{d t}$$ So $$M\frac{d \mathbf{v}}{d t} = M\frac{d \mathbf{v}}{d t} - \mathbf{u}\frac{d M}{d t}$$ And $$-...
  9. Sciencemaster

    I Does the wave function spread more quickly after it is observed?

    For the sake of this question, I am primarily concerned with the position wave function. So, from my understanding, the wave function seems to 'collapse' to a few states apon measurement. We know this because, if the same particle is measured again shortly after this, it will generally remain in...
  10. K

    I Electron angular momentum in diatomic molecules

    Hello! I just started reading some molecular physics and I am a bit confused about the electron angular momentum in diatomic molecules. Let's say we have just 2 protons and an electron for simplicity and we are in the Born-Oppenheimer approximation, so we assume that the nuclei are fixed in...
  11. Diracobama2181

    Energy Momentum Tensor in Phi^3 Theory

    $$ \bra{ \vec{ p'}} T_{\mu,\nu} \ket{ \vec {p}}=\bra{\Omega}\hat{a}(\vec{p'})(\partial^{\mu}\Phi\partial^{\nu} \Phi-g^{\mu \nu}\mathcal{L})\hat{a}^{\dagger}(\vec{p})\ket{\Omega}=\bra{\Omega}(\hat{a}(\vec{p'})\partial^{\mu}\Phi\partial^{\nu} \Phi\hat{a}^{\dagger}(\vec{p})\\...
  12. A

    Exploring Angular Momentum: Examining Earth & Bike Wheels

    Take for example earth. Earth has angular momentum about its own axis. However, if we ignore the orbital portion, the angular momentum of the Earth relative to the sun's axis is the same. Another example is the spinning bike wheel/person holding it in a chair. It has angular momentum about its...
  13. P

    Is the latter part of the momentum conservation question correct?

  14. jjson775

    Energy - momentum relationship

    The textbook says that by squaring and subtracting the expressions you can eliminate u. E² - p² = y² (mc²)² - y²m²u²
  15. Kaguro

    Fractional increase of energy vs momentum with relativity

    My attempt: ##E^2 = p^2c^2 + m_0^2c^4## ##2E dE = 2pc^2 dp ## ##\frac{dE}{E} = \frac{pc^2}{E^2}dp=\frac{p^2c^2}{E^2}## % (dp/p = 1%) ##=\frac{E^2-m_0^2c^4}{E^2}## % ##=1-\frac{m_0^2c^4}{E^2}## % ##=1-\frac{m_0^2c^4}{m^2c^4}## % ##=1-\frac{1}{\gamma ^2}## % ##=\frac{v^2}{c^2} ##% =0.81...
  16. Matejxx1

    Describe the motion of yoyos suspended from the ceiling

    I have trouble solving this problem any help would be appreciated.Problem statement ##J=\frac{mr^2}{2}## a) Determine the motion of yoyos for ##n=1,2,3## The case for ##n=1## is simple, however, I am having trouble with ##n=2## and ##n=3##. for ##n=2## I started by drawing all the forces...
  17. C

    Solving for $$\omega_2$$ using Conservation of Angular Momentum

    Unfortunately, I couldn't arrive to the correct answer ($$=0.28mL^2 \omega^2$$ ) and will be happy to understand what am I doing wrong. **My attempt:** Using $$ E_k = \frac{1}{2} I \omega^2 $$ I obtain that the difference I need to calculate is $$ \frac{1}{2} (2mL^2)(0.8\omega)^2 +...
  18. A

    Spinning Bike Wheel Example, how is angular momentum conserved?

    In the classic example of a person holding a spinning bike wheel, as they flip the wheel over, angular momentum is conserved by the person/chair spinning with 2x the angular momentum of the initial wheel. Not questioning that. However, I thought ang momentum is always conserved about a...
  19. B

    The definition of generalised momentum

    Why, in lagrangian mechanics, do we calculate: ##\frac{d}{dt}\frac{\partial T}{\partial \dot{q}}## to get the (generalised) momentum change in time instead of ##\frac{d T}{dq}##? (T - kinetic energy; q - generalised coordinate; p - generalised momentum; for simplicity I assumed that no external...
  20. mattlfang

    A ball hitting a two-ball system (with a spring between them)

    I honeslty don't quite know how to start. It seems like the Hooke's coefficent k is independent of the answer to this problem. I would also appreciate any clue of expressing the condition when "balls will collide again". The fact that all balls can keep moving make this rather difficult. It...
  21. Kaguro

    Angular momentum of orbit from orbit parameters and mass of sun

    L = mvr = mr (dr/dt) = 2m*r*(dr/dt)/2 = 2m*(dA/dt) So, A = (L/2m)T so, ## L = \frac{2 \pi a b m}{T}## Now, ##T^2 = \frac{4 \pi^2}{GM} a^3## So from all these, I get ##L = \sqrt{ \frac{GM m^2 b^2}{a}}## But answer given is ##L = \sqrt{ \frac{2GM m^2 ab}{a+b}}## (This, they have derived from...
  22. Kaguro

    Can spin angular momentum get converted to orbital angular momentum?

    I know that in QM, there is LS coupling. So the interaction is there. But is such an interaction possible in macroscopic objects like a planet?
  23. E

    I Relativistic Velocity, Perp. Accel., Momentum: Explained

    A stationary observer sees a particle moving north at velocity v very close to the speed of light. Then the observer accelerates eastward to velocity v. What is its new total velocity of the particle toward the north-west relative to the observer? I ask because while the particles total...
  24. Leo Liu

    Why is angular momentum measured at a non inertial CoM conserved?

    Question 7.6 Official solution It seems that the solution uses the conservation of angular momentum to solve this question (τ=0). But the problem is that the frame is set on the centre of mass of the guy, which is non inertial. I would like to know why it is correct to do it this way. My...
  25. T

    Why do bar magnets have zero velocity after collision?

    I don't understand the reasoning of this question's answer. The answer is velocity = 0 (option A). A while ago, I was told that, since the magnets were held at-rest (before being let go), they must have no velocity after the collision. What about the velocity which they had just before the...
  26. B

    2 contradicting approaches for a 1D elastic collision

    So I've managed to confuse myself on this problem :) Since the problem says we can assume ##m_p << m_b##, I'm assuming that the velocity of the bowling ball will be unchanged, such that ##\vec v_{b,i} = \vec v_{b,f} = -v_{b,0} \hat i## I started out using the energy-momentum principle, ##(\vec...
  27. Ayandas1246

    Conceptual questions about Angular Momentum Conservation and torque

    List of relevant equations: Angular Momentum = L (vector) = r(vector) x p(vector) Angular velocity of rotating object = w(vector), direction found using right hand rule. Torque = T(vector) = dL(vector)/dt I have a few questions about torque and angular momentum direction and...
  28. Leo Liu

    Is angular momentum conserved in a non-inertial frame?

    Question: If we place the frame of reference on an accelerating point, does the total rotational momentum still remain the same? I attempted to solve this question by manipulating the equations as shown below. $$\text{Define that }\vec r_i=\vec R+\vec r_i'\text{, where r is the position vector...
  29. J

    Angular Momentum: Spinning Mass

    First I calculated the momentum of m1. Since m2 was at rest after the collision, all its momentum was transferred, so m1 has a momentum of 158 i hat. L=r x p, so its 916 k hat. This would also be the change in L because it was initially 0 when m1 had no velocity, so I know this is the net...
  30. J

    Angular Momentum Problem: Rotation Rate

    First I found the moment of inertia, I=1.8(5.5^2+3.9^2+4.9^2) =125.046 Then I tried to find the rotation rate using the equation L=rotation rate*I rotation rate=3773/125.046=30.173 But the answer is suppose to be 21.263?
  31. J

    Angular Momentum Problem: Torque after fall

    I know how to get to the answer but that's what is confusing me. To find final velocity I multiply the acceleration by the time the object fell. Then multiply the velocity by the mass to get momentum. Now the angular momentum is r x p. Since the initial angular momentum was 0, this was also...
  32. J

    Angular Momentum Problem: Determining Angular Acceleration

    So I first tried to find L using torque, Torque=d/dt*L, and took the integral of this. Ended up with 23.28484t Now I square the equation L=rotation rate*I to get L^2=rotation acceleration *I^2 Angular acceleration=L^2/I^2 I feel like I am doing something wrong though, this doesn't give the...
  33. richengle

    Inelastic collisions -- how is momentum conserved but not energy?

    m1v1+m2v2=m1vs'+m2v2' , if car hits small fluffy object m2, initially v2=0, and v1'=v2' ... so m1v1=[m1+m2](v2') but why not energy? Why is there a KElost? .5m1v1^2+.5m2v2^2=.5m1v1'^2+.5m2v'2^2 +KElost , and again v2=0, v1'=v2' .5m1v1^2=.5[m1+m2]v2'^2+KElost using consv of momentum...
  34. OscarF

    Calculate speed from elastic and inelastic collisions? (momentum)

    So to cut to the chase, I missed my class' lesson on momentum - have tried to catch up, quite successfully but am baffled about this question. I know the conservation of momentum etc. but after trying for ages it's just not happening this question so any help would be much appreciated, Oscar.
  35. T

    Ball hitting racket - Momentum Question (ENGAA 2017)

    Please scroll-sown to Question 52: https://www.undergraduate.study.cam.ac.uk/files/publications/engineering_s1_qp_2017.pdf The correct answer is 'B'. This is the working I did: F = (change in momentum) / (change in time) change in momentum = mv - mu, where v = final velocity and u = initial...
  36. M

    Conservation of Angular Momentum -- Child jumping onto a Merry-Go-Round

    So we know that the initial intertia of the merry go round is 250 kg m^2 and its angular speed is 10 rpm. MGRs angular momentum would be L=Iw=250(10)=2500kg m^2 rpm. We know the mass if the child is 25kg, and the child's linear velocity is 6m/s. We convert linear to angular w= v/r = 6/2 =...
  37. WonderKitten

    Conservation of angular momentum

    Hi, I have the following problem: A homogeneous disc with M = 1.78 kg and R = 0.547 m is lying down at rest on a perfectly polished surface. The disc is kept in place by an axis O although it can turn freely around it. A particle with m = 0.311 kg and v = 103 m/s, normal to the disc's surface at...
  38. Saptarshi Sarkar

    Conservation of angular momentum under central forces

    I know that the force must be a central force and that under central forces, angular momentum is conserved. But I am unable to mathematically show if the angular and linear momentum are constants. Radial Momentum ##p=m\dot r = ma\dot \theta=ma\omega## Angular Momentum ##L=mr^2\dot\theta =...
  39. F

    B Where Does Momentum Go Near Merging Black Holes?

    Let's say you start 2 rockets in the opposite direction from a platform that's close to two soon to be merging black holes, the first rocket starts way before the second, but the second will ultimately fly a bit further, before they stop and fly back to towards the black hole(s). When black...
  40. JD_PM

    Dirac-Hamiltonian, Angular Momentum commutator

    We want to show that ##[\hat{ \vec H}, \hat{ \vec L}_T]=0##. I made a guess: we know that ##[\hat{ \vec H}, \hat{ \vec L}_T]=[\hat{ \vec H}, \hat{ \vec L}] + \frac 1 2 [\hat{ \vec H}, \vec \sigma]=0## must hold. I have already shown that $$[\hat{ \vec H}, -i \vec r \times \vec \nabla]= -...
  41. B

    B Does Jupiter have an angular momentum problem?

    I'm sure I've read somewhere that Jupiter has 99% of the solar system's angular momentum, which shouldn't be the case. However, I can't find a source for this, and any search online for the topic doesn't bring up any science sites. Did I mis-remember?
  42. K

    What determines that a system carries no angular momentum?

  43. S

    Why do heavier projectiles tend to have higher momentum?

    I am wondering why heavier bullets have a higher momentum than lighter bullets when using similar powder charges? At the muzzle, a typical 150 grain bullet fired from a 30-'06 will travel at around 3000 ft/s. A 200 grain bullet from the same rifle will travel around 2600 ft/s. (The velocities...
  44. SilverSoldier

    Collisions, Impulses and Impulsive Tension

    1. When an object attached to a fixed point with a string, is given a velocity and the string goes taut. So it says in this book (Applied Mathematics 1 by L. Bostock and S. Chandler) that when the string goes taut, the component of the velocity of the particle becomes zero in the direction...
  45. T

    Clarification on angular momentum

    I am confused with the following two questions: 1. A particle moves under the influence of a central force directed toward a fixed origin ##O##. Explain why the particle's angular momentum about ##O## is constant. 2. Consider a planet orbiting the fixed sun. Take the plane of the planet's...
  46. A

    EENGA 2019 Momentum question -- collision of two masses

  47. A

    Why Does Kinetic Energy Change After Collision in EENGA 2019 Question?

    A particle of mass m has kinetic energy E when it collides with a stationary particle of mass M. The two particles coalesce. Which of the following expressions gives the total kinetic energy transferred to other forms of energy in the collision? I keep getting C as my answer when the correct...
  48. B

    Momentum Thrust of an Over-Expanded Rocket Nozzle

    Hi All, I'm trying to get a better understanding of the momentum thrust given by an over-expanded rocket nozzle (I realize this case voids the isentropic flow assumption used for the 1D isentropic gas expansion equations typically used for rocket engine design since the normal shock is not an...
  49. JD_PM

    Evaluating a momentum operator

    I think I get the approach. We first need to evaluate the term ##\dot A_{\mu} \nabla A^{\mu}## and then evaluate the 3D space integral; we may need to take the limit ##V \rightarrow \infty## (i.e ##\sum_{\vec k} (2 \pi)^3/V \rightarrow \int d^3 \vec k##) at some point. The mode expansions of...
  50. E

    What are angular momentum and torque?

    Wikipedia says that they are the equivalents of momentum and force in rotational motion but I don't understand why this comparison is possible. The torque's dimension is N*m it seems like energy. What is this energy? Why angular momentum is not mass times angular velocity?
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