Hurricanes can involve winds in excess of 120 km/h at the outer edge.
Make a crude estimate of the energy of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 kg*m^3 of radius 120 km and height 4.5 km.
Make a crude estimate of the angular momentum of such a hurricane.
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To estimate the energy of a hurricane, we can assume it as a rigidly rotating uniform cylinder of air and use its kinetic energy formula. The kinetic energy of an object can be calculated using the formula:
KE = 0.5 * I * ω^2
Where:
KE is the kinetic energy
I is the moment of inertia
ω is the angular velocity
To estimate the angular momentum of a hurricane, we can use the concept of conservation of angular momentum. The angular momentum of an object is given by the formula:
L = I * ω
Where:
L is the angular momentum
I is the moment of inertia
ω is the angular velocity
Let's calculate both the energy and angular momentum of the hurricane using these formulas.
Step 1: Calculate the moment of inertia (I)
The moment of inertia of a cylinder can be calculated using the formula:
I = 0.5 * M * R^2
Where:
M is the mass of the cylinder (density * volume)
R is the radius of the cylinder
To find the mass (M), we can calculate the volume of the cylinder:
Volume = π * radius^2 * height
Given:
Density (ρ) = 1.3 kg/m^3
Radius (R) = 120 km = 120,000 m
Height (h) = 4.5 km = 4,500 m
Volume = π * (120,000)^2 * 4,500 ≈ 2.03 × 10^15 m^3
M = ρ * Volume ≈ 1.3 kg/m^3 * 2.03 × 10^15 m^3 ≈ 2.63 × 10^15 kg
Now, let's calculate the moment of inertia:
I = 0.5 * M * R^2 ≈ 0.5 * 2.63 × 10^15 kg * (120,000 m)^2 ≈ 1.798 × 10^28 kg*m^2
Step 2: Calculate the angular velocity (ω)
To estimate the angular velocity, we can use the assumption that the outer edge of the hurricane has winds exceeding 120 km/h. Let's convert this to m/s:
Speed = 120 km/h = 120,000 m/3,600 s ≈ 33.3 m/s
The angular velocity can be calculated using the formula:
ω = Speed / Radius
ω = 33.3 m/s / 120,000 m ≈ 2.775 × 10^(-4) rad/s
Step 3: Calculate the kinetic energy (KE)
Now that we have the moment of inertia (I) and angular velocity (ω), we can calculate the kinetic energy using the formula:
KE = 0.5 * I * ω^2
KE = 0.5 * (1.798 × 10^28 kg*m^2) * (2.775 × 10^(-4) rad/s)^2 ≈ 7.419 × 10^19 J
Therefore, the estimated energy of such a hurricane is approximately 7.419 × 10^19 Joules.
Step 4: Calculate the angular momentum (L)
To calculate the angular momentum, we can use the formula:
L = I * ω
L = (1.798 × 10^28 kg*m^2) * (2.775 × 10^(-4) rad/s) ≈ 4.987 × 10^23 kg*m^2/s
Therefore, the estimated angular momentum of such a hurricane is approximately 4.987 × 10^23 kg*m^2/s.