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 calculate its kinetic energy, assuming it is a rigidly rotating uniform cylinder of air. The formula for the kinetic energy of a rotating object is:
KE = 0.5 * I * ω^2
Where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
First, let's find the volume of the cylinder:
Volume = π * r^2 * h
Where r is the radius and h is the height of the cylinder.
Volume = π * (120,000 m)^2 * 4,500 m = 2.595e+13 m^3
Next, we need to find the mass of the air in the cylinder:
Mass = Volume * density
Mass = 2.595e+13 m^3 * 1.3 kg/m^3 = 3.3735e+13 kg
Now, let's estimate the angular velocity of the hurricane. The hurricane is rotating at 120 km/h, which is approximately 33.33 m/s.
Angular velocity = (2π * r) / time
Assuming a 120 km radius (120,000 m) and 33.33 m/s:
Angular velocity = (2 * 3.14 * 120,000 m) / 33.33 s ≈ 2,263 rad/s
To calculate the moment of inertia, we will use the formula for the moment of inertia of a cylinder:
I = 0.5 * m * r^2
Moment of inertia = 0.5 * 3.3735e+13 kg * (120,000 m)^2 = 5.119e+25 kg m^2
Finally, let's calculate the kinetic energy of the hurricane:
KE = 0.5 * (5.119e+25 kg m^2) * (2,263 rad/s)^2 ≈ 1.692e+31 joules
So, the estimated energy of the hurricane is approximately 1.692e+31 joules.
Now let's estimate the angular momentum of the hurricane. The formula for angular momentum is:
L = I * ω
Angular momentum = 5.119e+25 kg m^2 * 2,263 rad/s ≈ 1.158e+29 kg m^2/s
So, the estimated angular momentum of the hurricane is approximately 1.158e+29 kg m^2/s.