Answer:
a. <em>v_r = 17.14 km/h</em>
b. <em>P_r - P_e = 1838.55 N/m²</em>
c. <em>h = 157.50 m</em>
d. <em>Air is compressible and does not meet the Bernoulli equation. There are more variables that change the behaviour of the air:</em>
- <em>Temperature is lower at higher heights</em>
- <em>Density varies with the temperature</em>
- <em>Pressure is lower at higher heights</em>
Explanation:
a. If angular momentum is constant: <em>(I_r)(ω_r) = (I_e)(ω_e)</em>
<em>Since ω = v/r and I = mr²</em>
Thus,
<em>(m(r_r)²)(v_r/r_r) = (m(r_e)²)(v_e/r_e) ⇒ (v_r)(r_r) = (v_e)(r_e)</em>
⇒ <em>v_r = ((v_e)(r_e))/r_r</em>
<em> = ((200)(30))/350</em>
<em> = 17.14 km/h</em>
b. v_e = 55.56 m/s v_r = 4.76 m/s
h_e = 0 h_r = 0 (Earth's surface)
By Bernoulli's equation:
<em>P_r + ρg(h_r) + 1/2(ρ)((v_r)²) = P_e + ρg(h_e) + 1/2(ρ)((v_e)²)</em>
<em>P_r - P_e = 1/2(ρ)((v_e)²- (v_r)²) = 1/2(1.20)(55.56² - 4.76²) </em>
<em> = 1835 N/m²</em>
Since P_r - P_e > 0 ⇒P_r > P_e (pressure is higher in the rim)
c. <em>Kinetic Energy = Gravitational Potential Energy</em>
<em>1/2(m)(v_e)² = mgh</em>
⇒ <em>h = (v_e)²/2g</em>
<em> = (55.56)²/2(9.80)</em>
<em> = 157.50 m</em>
d. <em>Air is compressible and does not meet the Bernoulli equation. There are more variables that change the behaviour of the air:</em>
- <em>Temperature is lower at higher heights</em>
- <em>Density varies with the temperature</em>
- <em>Pressure is lower at higher heights</em>