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.
To estimate the energy of a hurricane, we can use the equation for the rotational kinetic energy of a rigid body:
E = (1/2) I ω²
where E is the energy, I is the moment of inertia, and ω is the angular velocity.
First, we need to calculate the moment of inertia of the hurricane, assuming it can be approximated as a uniform cylinder. The moment of inertia for a solid cylinder is given by:
I = (1/2) m r²
where m is the mass and r is the radius. We can calculate the mass by multiplying the density (ρ = 1.3 kg/m³) by the volume of the cylinder:
V = π r² h
where h is the height of the cylinder. So, the mass is:
m = ρ V = ρ π r² h
Now, let's calculate the moment of inertia:
I = (1/2) m r² = (1/2) ρ π r² h r²
Next, we need to estimate the angular velocity ω of the hurricane. We know the velocity of the winds at the outer edge is 120 km/h, which is equivalent to 33.3 m/s. For the approximation of a rigidly rotating cylinder, we can assume that the velocity is constant throughout, so the angular velocity would be:
ω = v / r
where v is the linear velocity of the winds at the outer edge and r is the radius. Therefore:
ω = 33.3 m/s / 120,000 m = 0.0002778 rad/s
Now, we can calculate the energy of the hurricane:
E = (1/2) I ω² = (1/2) (1.3 kg/m³) (π) (120,000 m)² (4,500 m) (120,000 m)² (0.0002778 rad/s)²
After evaluating this expression, we can find the approximate energy of the hurricane.
To estimate the angular momentum of the hurricane, we can use the formula:
L = Iω
where L is the angular momentum.
Using the previously calculated values for I and ω, we can compute:
L = (1/2) (1.3 kg/m³) (π) (120,000 m)² (4,500 m) (120,000 m)² (0.0002778 rad/s)