Answer to Question #181965 in Classical Mechanics for Robin

Question

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.

Answer 1

1. Given distance between the wall$(L)=\frac{n\lambda}{2}$

$\lambda = 2L =2\times 2.55 m= 5.10m$

2.

$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$

Now, substituting the values,

$P_2=\frac{P_1V_1\times T_2}{T_1\times V_2}$

$=\frac{1\times 10^5\times 1\times 10^{-3}\times (273-53)\times 2}{(273+20)\times 10^{-3}}$

$=1.5\times 10^5 atm$

4.

Now, applying the conservation of energy,

$\frac{1}{2} mv^2=eV$

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

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

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

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

5.

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

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

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

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

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

$v_{avg}=\frac{d\overrightarrow{r}}{dt}$

$=\frac{1}{x}f(\frac{t}{x})\hat{i}+\frac{2}{x}g(\frac{2t}{x})\hat{j}+\frac{3}{x}h(\frac{3t}{x})\hat{k}$