I'm a bit confused about how to solve a problem in which these two conservative quantities appear at the same time. Say for example, a pendulum with mass being hit by a particle with mass on its free spinning end.  I understand that both linear momentum and angular momentum are conservative, and know how to apply that to solve a variety of only-linear or only-angular problems, but i don't seem to understand what happens when you have to take on account both at the same time, is there a link between the mv and Iw equations I'm missing?
57828
I am trying to understand conservation linear momentum with in angular momentum  Herewith i just explain what i am  looking for.    Imagine an object of mass(M), velocity(V), is undergoing circular motion    with radius(R)     Angular momentum L = RMV  (R-radius M-mass- V- tangential velocity)  Mass is constant here  Since L is conserved   so when R is decreased tangential velocity(V)     has to be increased to keep angular momentum constant.  Does this mean linear momentum not conserved when radius of rotation changes?  When linear momentum is not conserved then what is the force that increases the velocity of object when R decreases.  Could anyone clarify this. or could any one say is my basic understanding about momentum is wrong. for easy understanding i am only considered magnitude of momentum's not cross product of vectors
38251
Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose because I was thinking how could you describe the motion of pendulum with an oscillating suspension point, something like this:  If you take your coordinate frame fixed in the suspension point, you would get the motion of a simple pendulum, which is false (because it's a non-inertial frame and you should have virtual forces). But if it were moving at constant speed, a moving coordinate frame should give the same equations of motion. So where does the restriction to non-inertial frames come from in the Lagrangian formalism? I've read that in non-inertial frames the lagrangian would be $L(q,\dot q, \ddot q, \dddot q..., t)$, but I didn't see a detailed explanation. And I'm not sure if the situation would be similar to Kepler's problem in polar coordinates.
97127
What is the underlying reason that the same pairs of conjugate variables (e.g. energy &amp; time, momentum &amp; position) are related in  (e.g. time symmetry implies energy conservation) and likewise in QM (e.g. $\Delta E \Delta t \ge \hbar$)?
63366
If we call the velocity of the mass $u$, then we can say: $u \cos θ = v$ and hence $u= \frac v{\cos θ}$. But why can't we write  $v \cos θ= u$ and add the contribution from both of the strings: $u= \frac{2v}{\cos θ}$ ? Why is it only $u= \frac v{\cos θ}$? Please note that it is not a homework based  question. I just want someone to solve the doubt arising in my mind.
117040
So this Youtube video has been around for a while now  . Also this  may be related.  I just wanted to know if there is any standard hydrodynamic terms to explain it. Derek clearly states it is not about Bernoulli principle, as in this case the ball rotation is important.  Has any recent paper/videos discussed what happens if you do the same experiment with hydrophobic spheres? Can it be predicted with laminar flow theories or does it need turbulence?  Too many questions have been posed, but not so many responses on the internet yet.  Edit: I said Derek stated "it wasn't Coandă effect", yet what he says is that it is not "just Bernoulli's principle". Coandă effect is the typical hairdryer ping-pong ball levitation which is explained by Bernoulli's principle.   Edit2: rephrasing.  Additional comment: There seems to be two explanations "water adhere to the sphere and is ejected conserving horizontal momentum" and "Magnus effect" (ME). The later is not so obvious to me as the system is in equilibrium and there are two fluids, water and air, (does ME means air pushes the ball horizontally?).   Edit3: DESCRIPTION To avoid rot links and people that do not want to see the video, I'll try to explain the phenomena. You have a sphere and a water jet. The water jet is vertical (opposite to gravity), and touches the sphere almost tangentially. The ball not only rotates due to the water jet but also stays at constant height from the ground. The angular velocity can be defined in the direction of the cross product between the position of the contact point between the jet and the sphere (origin in the center of the sphere) and the direction of the jet. Some water drops fly away at the top of the sphere away from the jet.  Edit4: New question, would this phenomenon occur if there was no air?
52873
thanks for any help.  I'm trying to show that in a 2body problem, angular momentum is conserved given that $\dfrac{dp}{dt}=\dfrac{-GMm(rv)}{r³}$, where p is momentum, t time, G gravitational constant, M mass of 1 object, m mass of the other, (rv) is the vector between them and r is the magnitude of the vector between them.  I've had a lot of attempts but I don't seem to get very far.  Where L is angular momentum, $r_1$ is position vector of mass M and $r_2$ is position vector of mass m, $p_1$ is momentum of mass M and $p_2$ is momentum of mass m.  L=$r_1Xp_1+r_2Xp_2$  If $\dfrac{dL}{dt}=0$ then dL will be conserved over time, applying product rule             $L=\dfrac{dr_1}{dt}Xp_1+r_1X\dfrac{dp_1}{dt}+\dfrac{dr_2}{dt}Xp_2+r_2X\dfrac{dp_2}{dt}$  noticing that $p_1=m\dfrac{dr_1}{dt}$ and $p_2=m\dfrac{dr_2}{dt}$ clearly $p_1$ is parallel to$ \dfrac{dr_1}{dt}$ and $p_2$ is parallel to $\dfrac{dr_2}{dt}$ and hence when cross producted with them, the result is 0.  Giving $L=r_1X\dfrac{dp_1}{dt}+r_2X\dfrac{dp_2}{dt}$  I gather I must now show that either these two terms are 0, or they are equal and     opposite. I can't see any reason why they would be 0, and (slightly guessing) it seems to    make sense to me that they are terms related to how the angular momentum of one body changes    with the other body, and I assume they must therefore be equal and opposite. However when I    do  $r_1X\dfrac{dp_1}{dt}=-r_2X\dfrac{dp_2}{dt}$      I notice $\dfrac{dp_1}{dt}=\dfrac{-GMm(rv)}{r³}$ and $\dfrac{dp_2}{dt}=\dfrac{GMm(rv)}{r³}$ (since (rv)'s direction is flipped)     Therefore $r_1X\dfrac{-GMm(rv)}{r³}=-\dfrac{r_2XGMm(rv)}{r³}$     Defining $w=\dfrac{GMm}{r³}$:  $r_1X(rv)(-w)=r_2X(rv)(-w)$     Scalar products are commutative so the (-w)'s easily cancel giving  $r_1X(rv)=r_2X(rv)$  Which then the only solutions are if all components of $r_1$ are equal to all components of      $r_2$, however angular momentum should clearly be conserved in all positions, not just when the masses are ontop of each other.  Thanks again for any help.
47850
We know that angular momentum is defined as the cross product of position and linear momentum. By taking the time derivative, we can deduce that the time rate of change of the angular momentum equals the net torque.  Then why is the conservation of angular momentum considered a law, when we can easily show it is conserved when no net torque is applied, by using Newton's laws?  It truly confuses me.
35511
If I understand correctly, spin is an intrinsic property of particles, which follows the algebra of angular momentum, but has nothing to do with an "orbital angular momentum" in that the particle is not like a small sphere that rotates on itself, to which we could attribute an angular momentum as usual in classical mechanics, cf.  and  Phys.SE posts.   My question is: why does spin combine with orbital angular momentum to give total angular momentum? To me this is surprising, as spin, in fact, has nothing to do with a classical angular momentum. Any insight on that?
68074
I understand the equations of angular motion and how they relate to one another, however it is not clear to me where the torque equation comes from.   We have $F=ma$. By solving for the angular interpretation of linear acceleration, we have $$a=\frac{dv}{dt}r=\frac{d\omega}{dt}r=\alpha r$$ and thus $F=m\alpha r$.   With $\tau=Fr$, $\tau=m\alpha r^2$. When we have multiple torques we can add them up because $\tau$ is linearly related to force and force is a sum. By defining $I=\sum{m_ir_i^2}$, we have $\tau=I\alpha$  I understand that the concept of inertia is really an abstraction. We see the discrete $\sum{m_ir_i^2}$ or $\int{m_ir_i^2}$ forms frequently when dealing with equations of angular dynamics and thus decide to name the quantity moment of inertia based on the common role it plays in the equations (eg: for kinetic energy)  My question, a clarification   However, I'm unclear about how we know $\tau=Fr$. Is $\overrightarrow{\tau}=rF$ a wholly mathematical corollary of newton's laws of motion, (without creating a tautology), or a later development backed by empirical research.
59688
Why is angular momentum conserved when a planet revolves about sun in an elliptical orbit? Why is linear momentum not conserved in this case?  Please use the minimum amount of equations and try to explain using pure physics logic if possible.
53153
I am currently stuck on the following question.   A stone is dropped from a stationary helicopter 500m above the ground, at the equator. How far from the point vertically below the helicopter does it land and in what direction? Solve this problem by conserving angular momentum.   My current interpretation is that when the stone falls in order to converse angular momentum the angular velocity must increase. This leads to the stone travelling to the east.   I equated the angular momentum at the start       L0=*(re+h)mVo  where re is radius of earth, h is the height above that, m is mass and Vo is inital velocity. To the angular momentum at any time  L=(re+h-1/2gt^2)m(Ve+Vo)  where the 1/2at^2 term comes from gravitational acceleration and Ve is the additional velocity.   I rearranged to find Ve and then integrated over the time it takes to fall the 500m in order to work out the extra distance the stone traveled. However, my answer is 1/2 what the answer should be. I have checked the maths and it seems to be fine. So I believe my set up is wrong. Any help would be appreciated.
109890
One simple problem is physics is to determine the mechanical energy difference after an explosion. To do this, you must assume that momentum is conserved because in a explosion you have internal forces, so, using Newton's third law, you get that the total momentum is conserved.  How true is that? I mean, when a explosion happens you have LIGHT and SOUND being propagated. And they carry energy and momentum. It's clear to me that if you only consider the momentum of the parts of the body + the momentum of the sound wave + the momentum of the light wave this should be equal to zero. However, it's not so clear if the momentum of the body should be equal of the sum of the momentum of its fragments. I have some conjectures, but I didn't find nothing related to that: I) The waves propagate symmetrically, so, because momentum is a vector, it should be zero. However, the explosion may not be total symmetric, so I don't know if this argument is valid. II) The momentum is not conserved for the body, but the momentum of the wave is negligible.   These are the two possibilities I considered, so I don't know if they are valid.
49433
So I have two particles that have collided in 3 dimensional space. I want the particles to rebound off of each other in an elastic manner. How do I determine the resultant velocity vector if I know: Both particles' initial velocity vector, their masses, and their initial position vectors relative to the origin.  I tried to follow the instructions on  but I am not sure how to make their equations work without polar coordinates / how to translate my data to polar coordinates.  Also, if you answer my question in terms of polar coordinates, can you explain which angle is theta and which is phi? Since there are 2 different conventions, I am not really sure which angle is being referred to in the explanation on the aforementioned website.  EDIT: This isn't for a class. I'm writing a physics engine for a gravity simulation I am making using Unity. I haven't taken kinematics since I was in high school, which is why I am so rusty.
14637
While trying to understand the second law of Newton from "An Introduction to Mechanics" by Kleppner and Kolenkow, I came across the following lines that I don't understand:      "It is natural to assume that for three-dimensional motion, force, like acceleration, behaves like a vector. Although this turns out to be the case, it is not obviously true. For instance, if mass were different in different directions, acceleration would not be parallel to force and force and acceleration could not be related by a simple vector equation. Although the concept of mass having different values in different directions might sound absurd, it is not impossible. In fact, physicists have carried out very sensitive tests on this hypothesis, without finding any variation. So, we can treat mass as a scalar, i.e. a simple number, and write $\vec{F} = m\vec{a}$."   The lines above lead me to question:   Why is it not" obviously true" that force behaves like a vector?  Why is it not impossible for mass values to be different in different directions?
69255
While studying rotational mechanics, I came across a section where it mentioned that angular momentum may not necessarily be parallel to angular velocity. My thoughts were as follows:  Angular momentum ($L$) has the relation $L=I\omega$ where $\omega$ is angular velocity and $I$ is the moment of inertia, so following this relation, it seems they should be in the same direction. Why are they not?
123462
For a long time I have wondered if there is a way to show that the rotational analogs of Newton's Laws are a direct consequence of just those laws, or are we adding more to them?  I understand mathematically that we take Newton's second law and do "r cross both sides," but that has always struck me as using more than just the 2nd law.  Here it matters where forces on the body are applied, but I don't see where Newton's Laws talk about points where forces are applied to a body. For translational motion it doesn't matter.  So are we adding a sort of "extra law" when we do this for rotation?  The only thing I can think of would be to "disassemble" an extended body into differential point masses and work with internal constraint forces that make the body rigid. If every mass is just a point mass it would get around the issue I'm having.  I've never seen any discussion along those lines, though.
29761
From :     "[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric."   What is the more detailed explanation for this, and/or where can I find it?