Momentum vs. Inertia

[Guille]

I have a good physics teacher, which is good at physics, but not explaining as a theacher. That is a really big problem. So, I have a physics exam in one wek and I want to understand something.....

What does momentum mean, is, and is calculated?

what does intertia mean, is, and is calculated?

Finally, what is the difference between momentum and inertia?


According to what I have understood from my teachers weak and deficult-to-follow explenations, in my head the significance of inertia and momentum is the same. is this correct?

[AntonioLao]

inertia is the property of matter which counteract any change to its linear momentum. Inertia is explicitly stated in Newton's 1st law of motion. Newton defined it as the inertial mass in contrast to gravitational mass. The inertial mass then appears in Newton's 2nd law of motion (F=ma), while the gravitational mass appears in Newton's universal law of gravitation (there are two disparate masses here: the very large mass creating the gravitational field and the tiny mass of the test point-particle in the field such as an apple). But Einstein formulated the Principle of Equivalence stating that the inertial mass is always equal to the gravitational mass. Nobody has yet come up with a good physical reason for this equality. Einstein used the Principle of Equivalence to formulate his general theory of relativity.

linear momentum (p) is the product of inertial mass and velocity or p=mv and since velocity is a vector, linear momentum is also a vector. The time rate of change of linear momentum is the inertial force (F=ma), which defines inertial acceleration as the time rate of change of velocity since the inertial mass is a constant for all practical purposes (at low energy and low speed). Newton's 3rd law of motion give assurance for the conservation of linear momentum.

When an object is approaching relativistic speed the linear momentum increases and is directly proportional to its speed.

the relativistic mass increase is given by special relativity as

[math]m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}[/math]

where [math]m_0[/math] is the rest mass.

The angular momentum is just the product of linear momentum and a metric. In quantum mechanics, the angular momentum is a constant equals to Planck's constant of action establishing Heisenberg's uncertainty principle as
[math]\Delta r \cdot \Delta p \geq \hbar[/math] where [math]\Delta r [/math] is the differential change in the metric and [math]\Delta p[/math] is the differential change in linear momentum.

[Guille]

One more thing, how is the momentum of an object calculated?

is mv^2 or something similar??? I think that's what they told me.........

[AntonioLao]

One more thing, how is the momentum of an object calculated?

From classical mechanics -
In an idealized frictionless system, the momentum of a point object of mass m and velocity v is given by mv. If there are no other object in the system, the velocity remains constant. But when interaction with one other object occurs then the vector sum of momentum remains constant. This is the conservation law for linear momentum and is a direct demonstration of Newton's 3rd law of motion.

[math]m_1 v_1 + m_2 v_2 = 0 [/math]

and the time rate of change of linear momentum is the inertial force.

[math] F_1 = m_1 \frac{d v_1}{d t}[/math]

[math] F_2 = m_2 \frac{d v_2}{d t}[/math]

therefore

[math] F_1 + F_2 = 0 [/math] or [math] F_1 = - F_2 [/math]

for every action there is an equal and opposite reaction.

But real objects are not point-particles, therefore in analytical mechanics, a center of mass is defined to account for linear momentum as well as angular momentum. Note that angular momentum is also conserved.

In relativistic mechanics, the time rate of change of mass is not zero, i.e., mass is not a constant.

[Guille]

In relativistic mechanics, the time rate of change of mass is not zero, i.e., mass is not a constant.

So, how is momentum calculated in relativistic mechanics?

[AntonioLao]

So, how is momentum calculated in relativistic mechanics?

the rest mass (m0) is replaced by the relativistic mass m_r given by

[math] m_r =\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}[/math]

so that the momentum becomes

[math] p = m_r \mathbf{v} [/math]

in experiments this is only noticeable when the velocity v approaches near the speed of light. The other problem is length contraction and time dilation.

[Guille]

the rest mass (m0) is replaced by the relativistic mass m_r given by

[math] m_r =\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}[/math]

so that the momentum becomes

[math] p = m_r mathbf{v} [/math]

in experiments this is only noticeable when the velocity v approaches near the speed of light. The other problem is length contraction and time dilation.

ok, I will memorize the Mr=......formula. and it is very obvious that it will only be noticeable when v is near c because it has to be a bick enough percentage to be devided by c and taken from 1, so that the number left can be devide rest mass, to get a relativistic mass.
What I mean is that I understand it.

[AntonioLao]

inertial mass of a 4D object is not noticeable until this 4D object moves in trying to cross into the 5th physical dimension. if the object succeeded in moving at the 5th dimension then it becomes a 1D object at higher LOE with constant velocity.

[Guille]

inertial mass of a 4D object is not noticeable until this 4D object moves in trying to cross into the 5th physical dimension. if the object succeeded in moving at the 5th dimension then it becomes a 1D object at higher LOE with constant velocity.

does this mean that when I move I'm trying to get into the fifth dimension?

also, when you say Energy is 3 dimensional, is it 2 spatial and time, or three spatial and no time? how can energy haver no time? when it's clear that, for example, heat can be changed from object (moved=motion=time) or KE of an object to another object (motion=time). energy does move. so it has time.

[AntonioLao]

does this mean that when I move I'm trying to get into the fifth dimension? also, when you say Energy is 3 dimensional, is it 2 spatial and time, or three spatial and no time? how can energy haver no time?

Yes. According to law of energy conservation, the total energy of the universe does not change and it is time independent. The time derivative of energy is always zero.

[math]\frac{\partial E}{\partial t} = 0 [/math]

But for Hamiltonian systems, the total energy is the sum of kinetic and potential energy but the potential energy is always found to be negative.