Question 199: Centre of Mass and Rotational Motion

Question

The mass of a hydrogen molecule is 3.32 $$	imes$$ 10
-27
kg. If 10
23
hydrogen molecules strike, per second, a fixed wall of area 2 cm
2
at an angle of 45
o
to the normal, and rebound elastically with a speed of 10
3
m/s, then the pressure on the wall is nearly:

Accepted Answer

Considering one hydrogen molecule :
As collision is elastic so, e = 1
Initial momentum,
$$\overrightarrow {{P_i}} $$ = $${{mv} \over {\sqrt 2 }}$$ $$\widehat i$$ $$-$$ $${{mv} \over {\sqrt 2 }}$$ $$\widehat j$$
Final  momentum,
$$\overrightarrow {{P_f}} $$ = $${{mv} \over {\sqrt 2 }}\left( { - \widehat i} ight)$$ $$-$$ $${{mv} \over {\sqrt 2 }}\widehat j$$
$$	herefore\,\,\,$$ Change in momentum for single H molecule,
$$\Delta $$P = $$\overrightarrow {{P_f}} $$ $$-$$ $$\overrightarrow {{P_i}} $$
= $${{2mv} \over {\sqrt 2 }}\left( { - \widehat i} ight)$$
$$	herefore\,\,\,$$ $$\left| {\Delta P} ight|$$ = $${{2mv} \over {\sqrt 2 }}$$
Now for n hydrogen molecule total momentum changes per second,
= $$\left( {{{2mv} \over {\sqrt 2 }}} ight)$$ $$ 	imes $$ n
As we know,
Force (F) = $${{\Delta P} \over {\Delta t}}$$
= $${{{2mv} \over {\sqrt 2 }}}$$ $$ 	imes $$ n
$$	herefore\,\,\,$$ As direction of $$\Delta $$P is towards negative i so force on the molecule will also be towards negative i direction. From Newton's third law, the reaction force will be on the wall in positive i direction with same magnitude.
$$	herefore\,\,\,$$ Force on the wall = $${{{2mv} \over {\sqrt 2 }}}$$ $$ 	imes $$ n
$$	herefore\,\,\,$$ Pressure on the wall,  P = $${F \over A}$$
= $${{2mv\,n} \over {\sqrt 2 A}}$$
= $${{2 	imes 3.32 	imes {{10}^{ - 27}} 	imes {{10}^3} 	imes {{10}^{23}}} \over {\sqrt 2  	imes2	imes {{10}^{ - 4}}}}$$
= 2.35 $$ 	imes $$ 10
3
N m
$$-$$2

Answer

2.35 $$	imes$$ 10
3
N m
-2

Answer

4.70 $$	imes$$ 10
3
N m
-2

Answer

2.35 $$	imes$$ 10
2
N m
-2

Answer

4.70 $$	imes$$ 10
2
N m
-2

Centre of Mass and Rotational Motion

The mass of a hydrogen molecule is 3.32 $\times$ 10 -27 kg. If 10 23 hydrogen molecules strike, per second, a fixed wall of area 2 cm 2 at an angle of 45 o to the normal, and rebound elastically with a speed of 10 3 m/s, then the pressure on the wall is nearly:

Choose the correct answer: