The Factors are Temperature and kinetic energy, the temperature is because the particles are going to move fast which means the particles in a solid container or in solid pattern they will actually vibrate and they will expand , when temperature increase, more kinetic energy between the particles .
examples is that in the steel when you heat a steel the particles inside it will vibrate then the particles speed up because the vibration increases, therefore the temperature increases so a a thermal expansion occurs that the vibration of the particles will take up more space so the steel bar expands slightly in all Direction if the temperature Falls the reverse happens and the material or steel contracts which means get smaller .
another example is the thermometer, the thermometer has a liquid inside it which is Mercury or alcohol this liquid expands when the temperature rises, the tube is made narrow so that a small increase in volume of the liquid produces a large movement along the tube.
Answer:
v = 7934.2 m/s
Explanation:
Here the total energy of the Asteroid and the Earth system will remains conserved
So we will have
![-\frac{GMm}{r} + \frac{1}{2}mv_0^2 = -\frac{GMm}{R} + \frac{1}{2}mv^2](https://tex.z-dn.net/?f=-%5Cfrac%7BGMm%7D%7Br%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv_0%5E2%20%3D%20-%5Cfrac%7BGMm%7D%7BR%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
now we know that
![v_0 = 660 m/s](https://tex.z-dn.net/?f=v_0%20%3D%20660%20m%2Fs)
![M = 5.98 \times 10^{24} kg](https://tex.z-dn.net/?f=M%20%3D%205.98%20%5Ctimes%2010%5E%7B24%7D%20kg)
![m = 5 \times 10^9 kg](https://tex.z-dn.net/?f=m%20%3D%205%20%5Ctimes%2010%5E9%20kg)
![r = 4 \times 10^9 m](https://tex.z-dn.net/?f=r%20%3D%204%20%5Ctimes%2010%5E9%20m)
![R = 6.37 \times 10^6 m](https://tex.z-dn.net/?f=R%20%3D%206.37%20%5Ctimes%2010%5E6%20m)
now from above formula
![GMm(\frac{1}{R} - \frac{1}{r}) + \frac{1}{2}mv_0^2 = \frac{1}{2}mv^2](https://tex.z-dn.net/?f=GMm%28%5Cfrac%7B1%7D%7BR%7D%20-%20%5Cfrac%7B1%7D%7Br%7D%29%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv_0%5E2%20%3D%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
now we have
![2GM(\frac{1}{R} - \frac{1}{r}) + v_0^2 = v^2](https://tex.z-dn.net/?f=2GM%28%5Cfrac%7B1%7D%7BR%7D%20-%20%5Cfrac%7B1%7D%7Br%7D%29%20%2B%20v_0%5E2%20%3D%20v%5E2)
now plug in all data
![2(6.67 \times 10^{-11})(5.98 \times 10^{24})(\frac{1}{6.37 \times 10^6} - \frac{1}{4 \times 10^9}) + (660)^2 = v^2](https://tex.z-dn.net/?f=2%286.67%20%5Ctimes%2010%5E%7B-11%7D%29%285.98%20%5Ctimes%2010%5E%7B24%7D%29%28%5Cfrac%7B1%7D%7B6.37%20%5Ctimes%2010%5E6%7D%20-%20%5Cfrac%7B1%7D%7B4%20%5Ctimes%2010%5E9%7D%29%20%2B%20%28660%29%5E2%20%3D%20v%5E2)
![v = 7934.2 m/s](https://tex.z-dn.net/?f=v%20%3D%207934.2%20m%2Fs)
Answer:
At 81. 52 Deg C its resistance will be 0.31 Ω.
Explanation:
The resistance of wire =![R_T =\frac{\rho_T \ l}{A}](https://tex.z-dn.net/?f=R_T%20%3D%5Cfrac%7B%5Crho_T%20%5C%20l%7D%7BA%7D)
Where
=Resistance of wire at Temperature T
= Resistivity at temperature T ![=\rho_0 \ [1 \ + \alpha\ (T-T_0\ )]](https://tex.z-dn.net/?f=%3D%5Crho_0%20%5C%20%5B1%20%5C%20%2B%20%5Calpha%5C%20%28T-T_0%5C%20%29%5D)
Where ![T_0 =20\ Deg\ C , \ \rho_0 = Constant, \alpha =3.9 \times 10^-^3 DegC^-1 \ (Given)](https://tex.z-dn.net/?f=T_0%20%3D20%5C%20Deg%5C%20C%20%2C%20%5C%20%20%5Crho_0%20%3D%20Constant%2C%20%20%5Calpha%20%3D3.9%20%5Ctimes%2010%5E-%5E3%20DegC%5E-1%20%5C%20%28Given%29)
l=Length of the wire
& A = Area of cross section of wire
For long and thin wire the resistance & resistivity relation will be as follows
![\frac{R_T_1}{R_T_2}=\frac{\rho_0(1+\alpha \cdot(T_1-2 0 )}{\rho_0(1+\alpha \cdot (T_2 -20 )}](https://tex.z-dn.net/?f=%5Cfrac%7BR_T_1%7D%7BR_T_2%7D%3D%5Cfrac%7B%5Crho_0%281%2B%5Calpha%20%5Ccdot%28T_1-2%200%20%29%7D%7B%5Crho_0%281%2B%5Calpha%20%5Ccdot%20%28T_2%20-20%20%29%7D)
![\frac{0.25}{0.31}=\frac{1}{[1+\alpha(T-20)]}](https://tex.z-dn.net/?f=%5Cfrac%7B0.25%7D%7B0.31%7D%3D%5Cfrac%7B1%7D%7B%5B1%2B%5Calpha%28T-20%29%5D%7D)
![1.24=1+\alpha (T-20)](https://tex.z-dn.net/?f=1.24%3D1%2B%5Calpha%20%28T-20%29)
![0.24=\alpha(\ T -20 )](https://tex.z-dn.net/?f=0.24%3D%5Calpha%28%5C%20T%20-20%20%29)
![Putting\ the\ value\ of \alpha = 3.9 \times 10^-^3 DegC^-1](https://tex.z-dn.net/?f=Putting%5C%20the%5C%20value%5C%20of%20%5Calpha%20%3D%203.9%20%5Ctimes%2010%5E-%5E3%20DegC%5E-1)
T = 81.52 Deg C
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the answer to your question!
Answer:
![y = 0.19 sin(5.23 t - 2.42x + \frac{\pi}{2})](https://tex.z-dn.net/?f=y%20%3D%200.19%20sin%285.23%20t%20-%202.42x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B2%7D%29)
Explanation:
As we know that the wave equation is given as
![y = A sin(\omega t - k x + \phi_0)](https://tex.z-dn.net/?f=y%20%3D%20A%20sin%28%5Comega%20t%20-%20k%20x%20%2B%20%5Cphi_0%29)
now we have
![A = 0.19 m](https://tex.z-dn.net/?f=A%20%3D%200.19%20m)
![\lambda = 2.6 m](https://tex.z-dn.net/?f=%5Clambda%20%3D%202.6%20m)
so we have
![k = \frac{2\pi}{\lambda}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B2%5Cpi%7D%7B%5Clambda%7D)
![k = \frac{2\pi}{2.6}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B2%5Cpi%7D%7B2.6%7D)
![k = 2.42 per m](https://tex.z-dn.net/?f=k%20%3D%202.42%20%20per%20m)
also we have
![T = 1.2 s](https://tex.z-dn.net/?f=T%20%3D%201.2%20s)
so we have
![\omega = \frac{2\pi}{T}](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Cfrac%7B2%5Cpi%7D%7BT%7D)
![\omega = \frac{2\pi}{1.2}](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Cfrac%7B2%5Cpi%7D%7B1.2%7D)
![\omega = 5.23 rad/s](https://tex.z-dn.net/?f=%5Comega%20%3D%205.23%20rad%2Fs)
now we know that at t = 0 and x = 0 wave is at y = 0.19 m
so we have
![\phi_0 = \frac{\pi}{2}](https://tex.z-dn.net/?f=%5Cphi_0%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D)
so we have
![y = 0.19 sin(5.23 t - 2.42x + \frac{\pi}{2})](https://tex.z-dn.net/?f=y%20%3D%200.19%20sin%285.23%20t%20-%202.42x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B2%7D%29)