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.
I'm looking into center of mass and I saw the derivation of:
## V = \frac{\sum\limits_{i = 1}^{n} m_iv_i}{\sum\limits_{i = 1}^{n} m_i} ##
I understand how it's derived, so no need to explain this further. It's a velocity of the frame in which total momentum of our objects is zero. Forget...
For this problem,
The reason why I am not sure whether it is a valid assumption whether momentum is conserved because during the collision if we consider the two masses to be the system, then there will be a uniform gravitational field acting on both masses, and a spring force that is acting...
From the bus driver's point of view, who is at rest, the ball's initial velocity is ##u+v##. After the collision, its velocity has to have the same value, but an opposite direction, so ##-(u+v)##. So that means that relative to me standing on the ground at rest, the ball's new velocity is...
I understand that conservation of motion comes from the action and reaction pairs of newton's third law. When it is triggered, two forces appear that cancel when analyzed as a system. My question is how is it that momentum is conserved if before the shot there was no force in the system and...
In the lab, how accurately can we measure momentum? What is the max value of the uncertainty in position as the uncertainty in momentum approaches zero? Or vice versa. What experiments do these types of measurements?
Hello, simplified the Angular momentum problem that comes up when i try to solve a mass moving inward or outwards and it does not conserver the angular momentum properly. I have tried this is many software by now, or by someone else and we all have found that there is no angular momentum...
The homework statement isn't exactly as is mentioned above. The actual problem statement is as follows:
This is problem 3.8 from John R. Taylor's Classical Mechanics; however, my question is not related to the main problem itself but one particular aspect of it. Now, in the same textbook (John...
I just don't understand should I take u relative to the plane or relative to the ground.
I tried to solve it like this:
$$p_{final}=m_{0}v-m(u-v)-M(u-v)$$
$$p_{initial}=m_{0}v$$
$$\Delta p=-m(u-v)-M(u-v)$$
##m_0## is mass of the plane.
$$F=\Delta p$$
$$F=-m(u-v)-M(u-v)=(m+M)(v-u)$$...
For this,
Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero?
Many thanks!
How should I calculate the angular momentum carried by a current carrying circular wire? Is it correct to consider the angular momentum of the electrons moving with drift velocity? Like
##L = n m_e v_{drift} r## where ##r## is radius of the loop, and ##n## is total number of electrons moving in...
I've already solved the orbital speed by equating the kinetic and potential energy in the circle orbit case.
$$\frac{1}{2}mv^2 = \frac{1}{2}ka^2.$$
And so $$v^2 = \frac{k}{m}a^2$$
Now when the impulse is added, the particle will obviously change course. If we set our reference point in time...
I'm guessing this question can be solved using the law of conservation of momentum
Vi = 5 m/s
(5 m/s) M = (4.33 m/s) cos30 M + V sinθ M
I don't know what to do after this... I'm also not sure if I use the sin and cos correctly.
A simultaneous measurement of both a particle's position and momentum may be successfully accomplished if more than one photon were utilized for the measurement. A non-demolishing measurement is possible if the emitters were aligned such that each would offset the other’s recoil of the target...
In Chapter 5.3, Ballentine uses geometrical arguments to obtain the initial magnitude of a hydrogen atom's bound electron momentum. How does equation (5.13) obtain? I tried to naively compute
$$p_e^2 \equiv \textbf{p}_e\cdot \textbf{p}_e = p_a^2+p_b^2+p_o^2 + 2\textbf{p}_a\cdot \textbf{p}_b -...
Hi
For an infinite well , solving the Schrodinger equation gives wavefunctions of the form sin(nπx/L). These are not eigenfunctions of the momentum operator which means there are no eigenvalues of the momentum operator. Does this mean momentum cannot be measured ?
Inside the infinite well the...
Hi,
Here is the problem
What is required to answer this question is two assumptions. Firstly, the component of the momentum normal to the centre line is the same before and after. Therefore, secondly, A must recoil entirely in the horizontal plane. This is the only way to answer this question...
I understand how a massive, electrically charged spinning ball would have both angular momentum and a magnetic dipole, and i can see how the
Stern–Gerlach experiment shows that the magnetic dipole of an electron is quantized.
What kind of experiment demonstrates
a connection between electron...
Why in order to derive the QM momentum operator we use the plane wave solution. Why later on in field theory and particle physics, the plane wave ansatz is so physically important?
Hi, I have a question.
Let us say we have the wave function as with eigen value and base eigen value of:
##!\psi >\:=\:\frac{1}{6}\left(4!1,0,0>\:+\:3!2,1,1>\:-1!2,1,0\:+\:\sqrt{10}!2,1,-1>\right)##
I need to find <Ly^2>
the solution of the problem according to answers, is demanding that...
Cohen tannoudji. Vol 1.pg 702
"Now, let us consider an infinitesimal rotation ##\mathscr{R}_{\mathbf{e}_z}(\mathrm{~d} \alpha)## about the ##O z## axis. Since the group law is conserved for infinitesimal rotations, the operator ##R_{\mathbf{e}_z}(\mathrm{~d} \alpha)## is necessarily of the...
I was playing around, and I found something unexpected. If we are analyzing a simple fluid jet:
We can apply Bernoulli's (which is Conservation of Energy) and arrive at:
$$ P_{1_{B}} = \frac{1}{2} \rho \left( v_2^2 - v_1^2 \right) = \frac{1}{2} \rho ( v_2 - v_1 ) ( v_2+ v_1) $$
It would...
I calculated:arctan(fy/13.0)=55=>fy=18.566 m/s
Then I calculated, using the momentum equation:
m1viy+m2v2iy=(m1+m2)vfy=>
mv2i=2*m*18.566=>v2=37.132 m/s
I thought that because the cars were stuck together, the kinetic energy from the northbound car would be lost. So, the speed would have...
I understand that it is a 2D momentum problem with an elastic collision;
Looking at the vector diagrams below, I notice that the velocity vectors initial and final in the y direction are in the same direction, indicating that momentum does not change, whereas the velocity vectors initial and...
I read that quantum spin is the measure of the angular momentum of a quantum object. Suppose you have a rotating Thing 1. Quantum objects bounce off of it then collide with Thing 2. Will this transfer angular momentum from Thing 1 to 2, causing it to rotate?
Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then
##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0##
will require that the momentum expectation value remains...
For this problem,
Why for part (a) the solution is,
Is the bit circled in red zero because since the putty is released at a very small distance above the rod it velocity is negligible?
Also for part (d) the solution is
I did a computation of the initial and finial kinetic energies of the...
Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
If so, What...
For this problem I was very confused whether conservation of angular momentum should be applied to the person, the swing or the person-swing system. It seems to me that there is no net torque on any of the three systems I listed above. However, it seems that the angular momentums of the three...
Sorry for this beginner's question, but...if F=ma, then force is all about acceleration. But if vehicle A moving at constant velocity V hits a wall, and vehicle B moving at constant velocity greater than V hits the wall, then B hits the wall with greater momentum than A and does greater damage...
Hi,
In my book I have and expression that I don't really understand.
Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt##
Where L...
Do you know of any place where I can look up things about the momentum (linear momentum) in fluid mechanics? It's just that when I have a variable velocity and it has to be integrated, I don't quite understand how to do it.
I have looked for videos and things and I can't find that they are...
Kinetic energy before collision =1/2 mv² + 1/2 mv² = mv² (since energy is a scalar quantity, the direction does not matter). Kindly tell why am I not getting the required answer i.e: 1/2 mv². Am I doing the calculation wrong?
^ This is my personal drawing of the diagram, I couldn't take a picture of the actual one. The setup is a pulley wrapped with a cord and mass hangers attached to each end.
My first thought when approaching this problem was to first determine the rotational inertia of the pulley, then use some...
My Explanation:
This system is a closed system, so the center of mass velocity stays constant. It was initially at rest so the position of the center of mass is constant. After their collision, the 2 carts are to the right of x = 0. Center of mass originally was at x = 0, so the platform had to...
The rotating ball should push the vehicle first to the right and once it hits the airbag - to the left?? Even if this works, how are you going to automate it and repeat it?
I have typed up the main problem in latex (see photo below)
It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.
The section Kepler’s Second Law here describes the above equation.
In this problem,
##\text{r = D, m = M and v = V}##
What is the way to go about finding out ##\theta## as shown in Figure 13.21?
When you write out the equations of motion for a system of two isolated charges, you can add both of the equations and get the increase in the particles linear momentum on one side. On the other side, you get the sum of all the forces between the particles. I understand that this sum of forces...
In relativity, momentum of a body is given by ##p=mv/\sqrt{1-v^2/c^2}##, but if mass is exactly zero and velocity is exactly ##c##, how is the photon momentum even defined? I don't think this problem can be resolved by simply stating the other formula relating energy to momentum, since it was...
So i was able to solve the angular velocity part but i don't know how to find the velocity of centre of mass . For the first part i simply conserved momentum about COM because if i consider the particles as a part of the same system as rod the collision are internal forces . I am mainly...
Specifically given a purely magnetic hamiltonian with some associated vector potential :
$$ H = \dfrac{1}{2m} (\vec{p} - q\vec{A}) $$
How can I deduce if $$ \vec{L} = \vec{r} \times \vec{p}$$ is conserved? ( $$\vec{p} = \dfrac{\partial L}{\partial x'}$$, i.e. the momentum is canonical)
[This is a continuation of OP's thread here: https://www.physicsforums.com/threads/satellite-mechanics-linear-and-rotational-momentum.1046963/ ]
satellite mechanics: linear and rotational momentum
I'm trying to better understand classical mechanics, and came up with a question:
Say we have a...
-Solved for vf using equation 3 to get 20.0m/s (speed before explosion) then solved for the distance to reach the explosion using equation 4, to get 20.0m, which felt wrong having the same numbers but that may just be coincidence.
-Found the distance travelled of the lighter piece using 530m -...
i,j,k arevector
I know L=P*r=m*v*r=m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k=mabωk.
but why m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k.I need some detail.
please help me.