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.
Angular momentum
= Iω kg-m²/s.
ω=angular velocity in rad./s.
I=moment of inertia
=(1/2)Mr²
M=total mass of air in kg.
r=outside radius of hurricane, in m.
To estimate the energy and angular momentum of a hurricane, we will approximate it as a rigidly rotating uniform cylinder of air.
First, let's calculate the volume of the cylinder. The volume of a cylinder is given by the formula:
V = πr^2h
where r is the radius and h is the height. Substituting the given values:
V = π * (120 km)^2 * 4.5 km
Convert the values to meters:
V = π * (120,000 m)^2 * 4,500 m
Now, let's calculate the mass of the air in the cylinder. The density of air is given as 1.3 kg/m^3, so we can calculate the mass using the formula:
m = ρV
where ρ is the density and V is the volume. Substituting the values:
m = 1.3 kg/m^3 * V
Now, let's calculate the energy of the hurricane. The kinetic energy of a rotating object is given by the formula:
E = (1/2)Iω^2
where I is the moment of inertia and ω is the angular velocity. In this case, we can approximate the moment of inertia of the cylinder as:
I = (1/2)mr^2
where m is the mass of the air in the cylinder and r is the radius. Substituting the values:
I = (1/2) * m * (120,000 m)^2
Now, let's estimate the angular velocity of the hurricane. The angular velocity of a rotating object is given by the formula:
ω = v / r
where v is the linear velocity and r is the radius. Since the wind speed at the outer edge of the hurricane is given as 120 km/h, we need to convert it to m/s:
v = 120 km/h * (1000 m/1 km) * (1 h/3600 s)
Substituting the values:
ω = v / (120,000 m)
Finally, let's substitute the calculated values into the energy formula to find the energy of the hurricane.
E = (1/2) * I * ω^2
Similarly, let's calculate the angular momentum, which is given by the formula:
L = I * ω
Now, with all the calculations explained, you can plug in the values into the respective formulas to obtain the crude estimate of the energy and angular momentum of the hurricane.