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.
After impact VCos(alpha) will be normal to the inclined, now i calculated for theta by using [ Tan(theta) = vCos(alpha)/vSin(alpha),. Tan(theta)= Cot(alpha) ]. I don't know how to solve further to get value of theta, according to book thta = alpha.
Now, the net vertical impulse on the wedge should be zero. It's quite obvious from the figure that the ground will also exert an impulse of ##J cos 30°## on the wedge. But they've given the answer as ##J sin 30°##.
They're wrong, right?
Solving using Linear Momentum:
M vb2/2 = M g 2L
vb = 2√(g L)
m v = m v/2 + M (2√(g L) )
v = 4 M √(g L) / m
Note: I see from the answers - that this is correct.
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Next, I tried to solve it via Energy conservation point of view.
M vb2/2 = M g 2L
vb = 2√(g L)
m v2/2 = m v2/8 + k...
Homework Statement
A system has a ball and a uniform rod. The rod is rotating about point X on a frictionless table until it strikes the ball. The rod stops and the ball moves away.
Variables:
Rod's mass: m1
Ball's mass: m2
Rod's original angular velocity: ω
Ball's final velocity: v
Rod's...
Homework Statement
A mass m_1, initially moving at a speed v_0, collides with and sticks to a spring attached to a second, initially stationary mass m_2. The two masses continue to move to the right on a frictionless surface as the length of the spring oscillates. At the instant that the...
Homework Statement
Four equal discs of mass ocuppy the vertices of a square frame made by four rigid bars of length and negligible mass. The frame is at rest on a horizontal table, and it can move with negligible friction. An instantaneous impulse is transmitted to one of the masses, in the...
Homework Statement
If we have a hollow ball completely filled with water which is rolling without slipping on a horizontal ground. If the water freezes which of the parameter will remain unchanged-
angular speed, angular momentum, linear momentum, kinetic energy, total energy
Homework...
Homework Statement
A 10 g bullet traveling at 400 m/s strikes a 10 kg , 1.2-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door immediately after impact?
Homework Equations
p[/B]= mv
L = Iω...
I am trying to create a momentum trap to calculate the velocity of a projectile when it hits the trap. It essentially consists of a plate if known mass at the bottom of a pendulum of known radius. When the projectile hits the plate, the degrees that the pendulum rotates are recorded. I believe...
Homework Statement
Homework Equations
Conservation of linear momentum
Conservation of angular momentum
##\frac{-v_{relative after collision}} {v_{relative before collision}}=e## where e is coefficient of restitution
The Attempt at a Solution
Components of velocity given as...
I understand for linear momentum the measure of the dificulty to take a body to the rest. But when I do the next exercises, I find that it requires less force for one body compared to the other with same linear momentum.
Can someone explain me what to understand in an intuitive way what to...
Hi all,
Yet another question: if the universe is finite, then linear momentum should be quantized (I assume in a similar manner to an infinite potential well since there are boundary conditions). My question is, then, if one computes a value for ##\Delta p## (momentum variance), is the variance...
Homework Statement
Let's have a cylinder in air tunnel. The air is flowing around cylinder. I want to calculate a net force acting on the cylinder. Part of the incoming air hits the cylinder and its trajectory direction changes. For that reason we can see straight behind the cylinder lower...
I've managed to derive the form of Reynolds transport theorem as a bilance of linear momentum of the system:
\left (\frac{\vec{\mathrm{d} p}}{\mathrm{d} \tau} \right )_{system}=\frac{\mathrm{d} }{\mathrm{d} x}(\int_{V}^{ }\vec{v}\cdot \rho dV)+\int_{A}^{ }\vec{a}dm+\int_{A}^{ }\vec{v}\cdot \rho...
So I did this lab on momentum on jets and the theory is that "a force is applied on the plate due to change of momentum" the jet both impacted on a flat plate and on a hemispherical plate; now i understand that the greater the angle of deflection of the water jet, the greater the change of...
Homework Statement
A firework of mass 1 kg is placed on the ground and ignited , the impulse created by the explosion causes it to move vertically upwards,after 4 seconds a second explosion takes place in air and the horizontal impulse forces cause it to separate into 4 identical pieces
2...
Hi I have been dealing with a fluid mechanics pressure gradient problem and from a statistical view point I can see how it resolves itself but am puzzled as to how it can occur at the molecular scale from a conservation of linear momentum perspective if Momentum is a conserved quantity
While...
Hi, I study linear momentum. But I cannot understand enough that how can forces appear on a pipe when fluid flows in the pipe? Fluid have a momentum as m*v how can forces appear on the pipe. I cannot demonstrate the situation in my mind. How can I find a demonstration or a free body diagram for...
Homework Statement
a small 0.199 kg block slides down a frictionless surface through height h = 0.608 m and then sticks to a uniform vertical rod of mass M = 0.398 kg and length d = 2.23 m. The rod pivots about point O through angle θ before momentarily stopping. Find θ.
point O is at the end...
1. Homework Statement
The problem is shown in the photo above. I would like to discuss part B.
Homework Equations
conservation of linear momentum
The Attempt at a Solution
the solutions say that we should have the following approach:
Mb*vb = [Mb + Ms] * vFinal.
Plug in the numbers and get...
Homework Statement
A railroad car moves under a grain elevator at a constant speed of 4.50 m/s. Grain drops into the car at the rate of 420 kg/min. What is the magnitude of the force needed to keep the car moving at constant speed if friction is negligible?
Homework Equations
U=V+dV-Vrel , U...
Homework Statement
Three blocks of identical mass are placed on a frictionless table as shown. The center block is at rest, whereas the other two blocks are moving directly towards it at identical speeds v. The center block is initially closer to the left block than the right one. All motion...
Referring to the Law of Conservation of Momentum: How is momentum always conserved? In a non-isolated system, an external force causes a change in momentum, so that initial momentum isn't the same as final momentum. Wouldn't this constitute a situation where momentum is not conserved?
Or, is...
I've been a fan of this forum for a while, but never signed up for it, today I'm stuck on this problem and can't find anywhere other than Chegg but I don't have a premium account.
1. Homework Statement
A meter stick is at rest on frictionless surface. A hockey puck is going towards the 30cm...
Ok, so as far as I understand it, it is impossible to turn linear momentum (p) into rotational momentum (L), but I don't quite understand why. The main thought experiment I have in my head is this:
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Homework Statement
Two identical buggies 1 and 2 of mass ##M## with one man of mass ##m## in each, move without friction due to inertia towards along two parallel rails. When the buggies are opposite to each other, the men exchange positions by jumping in a direction perpendicular to motion of...
Homework Statement
There are two trolleys traveling in the same direction which are about to collide.
Trolley 1 is traveling at 4 metres per second and has a mass of 2kg
The other, Trolley 2, (which is in front) is moving a 1 metre per second and has a mass of 4kg.
After the impact, they...
Consider a particle with velocity "v" has the collision with a rotating disc.
How can I analyze the final angular velocity this system?
If the mass of particle is very negligible related to the mass of rotating disc, definitely particle will turn back after collision. In this case, how can I...
Homework Statement
If an analysis of momentum and impulse, a rigid body has initial linear and angular momentum zero, does this mean that the vector sum of the impulses and the vector of LINEAR momentum final, will be in the same direction?
Homework Equations
Systema moment 1 + System Extern...
Homework Statement
Two atoms of equal mass m, that move with the same speed but opposite direction, interact when they're in some region R of space, as in fig.1. After the interaction, one of the atoms moves with velocity ## \vec{V1} ## as in fig.2.
a) Are the linear and angular momentum of...