Answer to Question #181965 in Classical Mechanics for Robin

Question #181965

1. a small bathroom has two very hard walls on opposite sides of the room at a distance of 2.55m.  these walls reflect sound so that sound waves have nodes at the walls.  for which approximate frequency can a standing wave occur between the walls (resonance).


2. a container holding 1 liter is filled with air of room temperature (20 degrees Celsius) and normal atmospheric pressure.  it is compressed to half volume and cooled to -53 celsius, what is the pressure in the air.


4. an electron is accelerated from the rest of a static electric field by a potential difference of 0.51 MV.  what speed do the electrons get?


5. a point particle moves in space so that its position vector is given by: ((x (t), y (t), z (t)) = (f (t / x), g (2t / x), h (  3t / x)) where f, g and h are three periodic functions with period 1 and x is a constant, what is the average of a velocity vector from time t to t + x.


1
Expert's answer
2021-04-19T17:10:55-0400
  1. Given distance between the wall(L)=nλ2(L)=\frac{n\lambda}{2}

λ=2L=2×2.55m=5.10m\lambda = 2L =2\times 2.55 m= 5.10m


2.


P1V1T1=P2V2T2\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}

Now, substituting the values,

P2=P1V1×T2T1×V2P_2=\frac{P_1V_1\times T_2}{T_1\times V_2}


=1×105×1×103×(27353)×2(273+20)×103=\frac{1\times 10^5\times 1\times 10^{-3}\times (273-53)\times 2}{(273+20)\times 10^{-3}}

=1.5×105atm=1.5\times 10^5 atm


4.

Now, applying the conservation of energy,

12mv2=eV\frac{1}{2} mv^2=eV

v=2eVm\Rightarrow v=\sqrt{\frac{2eV}{m}}

v=2×1.6×1019×0.51×1069.1×1031m/s\Rightarrow v = \sqrt{\frac{2\times 1.6\times 10^{-19}\times 0.51\times 10^{6}}{9.1\times 10^{-31}}} m/s


v=0.18×1018m/s\Rightarrow v =\sqrt{0.18\times 10^{18}}m/s


v=0.424×109m/s\Rightarrow v = 0.424\times 10^{9}m/s


5.


x(t)=f(tx)i^x(t)=f(\frac{t}{x})\hat{i}


y(t)=g(2tx)j^y(t)= g(\frac{2t}{x})\hat{j}


z(t)=h(3tx)k^z(t)=h(\frac{3t}{x})\hat{k}

r=x(t)+y(t)+z(t)\overrightarrow{r}=x(t)+y(t)+z(t)


=f(tx)i^+g(2tx)j^+h(3tx)k^=f(\frac{t}{x})\hat{i}+g(\frac{2t}{x})\hat{j}+h(\frac{3t}{x})\hat{k}


vavg=drdtv_{avg}=\frac{d\overrightarrow{r}}{dt}


=1xf(tx)i^+2xg(2tx)j^+3xh(3tx)k^=\frac{1}{x}f(\frac{t}{x})\hat{i}+\frac{2}{x}g(\frac{2t}{x})\hat{j}+\frac{3}{x}h(\frac{3t}{x})\hat{k}


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