The classical motion for an oscillator that starts from rest at location x₀ is
x(t) = x₀ cos(ωt)
The probability that the particle is at a particular x at a particular time t
is given by ρ(x, t) = δ(x − x(t)), and we can perform the temporal average
to get the spatial density. Our natural time scale for the averaging is a half
cycle, take t = 0 → π/
ω
Thus,
ρ = 
Limit is 0 to π/ω
We perform the change of variables to allow access to the δ, let y = x₀ cos(ωt) so that
ρ(x) = 
Limit is x₀ to -x₀
Limit is -x₀ to x₀
This has 
as expected. Here the limit is -x₀ to x₀
The expectation value is 0 when the ρ(x) is symmetric, x ρ(x) is asymmetric and the limits of integration are asymmetric.
Answer:
Work = 651,1011 kJ
Explanation:
Let´s take the car as a system in order to apply the first law of thermodynamics as follows:
Where
And considering that there is no mass transfer and that the only energy flows that interact with the system are the heat losses and the work needed to move the car we have:
Regarding the energy system we have the following:
By doing the calculations we have:
![E_{system,final}- E_{system,initial}=[0,1*900]_{internal}+[0,5*900(30^2-10^2)/1000)_{kinetic}+(900*10*(20-0)/1000)_{potential}\\E_{system,final}- E_{system,initial}=90+360+180=630kJ](https://tex.z-dn.net/?f=E_%7Bsystem%2Cfinal%7D-%20E_%7Bsystem%2Cinitial%7D%3D%5B0%2C1%2A900%5D_%7Binternal%7D%2B%5B0%2C5%2A900%2830%5E2-10%5E2%29%2F1000%29_%7Bkinetic%7D%2B%28900%2A10%2A%2820-0%29%2F1000%29_%7Bpotential%7D%5C%5CE_%7Bsystem%2Cfinal%7D-%20E_%7Bsystem%2Cinitial%7D%3D90%2B360%2B180%3D630kJ)
Consider that in the previous calculation, the kinetic and potential energy terms were divided by 1.000 to change the units from J to kJ.
Finally, the work needed to move the car under the required conditions is calculated as follows:
Consider that in the previous calculation, the heat loss was changed previously from BTU to kJ.
Translate in Spanish: lo siento, hubiera puesto 300 puntos pero los usé todos para mi última pregunta
If you meant a different language lmk
