15
votes
Sum of the individual kinetic energies of the particles which make the system the same as the K.E. of the center of mass? What's bad in my reasoning?
A simple example to show that the kinetic energy of the CoM is not the same as the sum of the kinetic energies of the particles which make up the system is the centre of mass being stationary and the ...
- 98.9k
7
votes
Sum of the individual kinetic energies of the particles which make the system the same as the K.E. of the center of mass? What's bad in my reasoning?
Sum of the individual kinetic energies of the particles which make the
system the same as the K.E. of the center of mass?
No.
The total kinetic energy consists of two parts.
One is the kinetic energy ...
- 76k
5
votes
Why are they integrating from infinity to $r$ in the derivation of GPE formula?
Not a dumb question at all. A general property of integrals is that when switching the integral bounds, there is just an overall minus sign:
$$\int_a^bf(x)dx = - \int_b^af(x)dx.$$
So, why do we even ...
- 181
4
votes
Sum of the individual kinetic energies of the particles which make the system the same as the K.E. of the center of mass? What's bad in my reasoning?
If we divide a body into multiple elemental masses $dm$ and each elemental mass of the body is having variable velocity $v$($v$ varies throughout the body) then the total kinetic energy is
$$K=\frac12\...
- 1,220
3
votes
Accepted
Helmholtz free energy density proof
Additivity with respect to extensive variables ($F(T, V_1 + V_2, N_1 + N_2 ) = F(T, V_1 , N_1 ) + F(T, V_2, N_2 )$ and continuity of the Helmholtz free energy imply it is a homogeneous function of ...
3
votes
Could the energy or more clearly energy transfer to a object be thought as force packets applied to it, like instances of force?
could the energy or more clearly energy transfer to a object be thought as force packets applied to it, like instances of force
No. Force is a transfer of momentum, not energy. So $\vec F=d\vec p/dt$....
- 105k
2
votes
Why are they integrating from infinity to $r$ in the derivation of GPE formula?
There is no rule that the upper limit should be larger than the lower limit of the integal. For the case of work, the choice of limits simply tells you for what motion is the work calculated. The ...
- 8,182
2
votes
Accepted
Sum of the individual kinetic energies of the particles which make the system the same as the K.E. of the center of mass? What's bad in my reasoning?
The flaw in your reasoning is that $dW = \sum F_i \cdot ds_i$. You're trying to factor out the $ds$ and say that $dW=\left( \sum F_i\right) ds$, but you can't factor out $ds$ because it's not the same ...
- 9,038
2
votes
Accepted
Trying to explain why $W = ΔKE$ and not $ΔE$
but for some reason, gravity did not remove any energy from the crate.
That’s not true. Gravity did negative work of $-mgh$ on the crate. The work is negative because the force of gravity is in the ...
- 76k
1
vote
Trying to explain why $W = ΔKE$ and not $ΔE$
Work is about energy changing from one form to another. For example, when you lift the crate, you convert chemical energy into gravitational potential energy.
If you drop the crate, you convert ...
- 42.8k
1
vote
What is meant by velocity along the time axis in relativity theory?
In relativity many things are unified which were originally considered to be separate. The most famous are space, $\vec x$, and time, $t$, which are unified to become spacetime, $(ct,\vec x)$.
Other ...
- 105k
1
vote
Sum of the individual kinetic energies of the particles which make the system the same as the K.E. of the center of mass? What's bad in my reasoning?
As @accumulation points out, the path in space over which the force is integrated is the problem.
(Just in case it is not clear) if the constituent parts are at rest in the center-of-mass system, the ...
- 96
1
vote
Does mass of comet change its orbit?
If it was a sphere and its mass were lost symmetrically (same speed) in all directions, it would not affect its orbit (Assuming the planet is large enough so its motion would not to be affected by ...
- 1,249
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