(a) Calculate the angular momentum of Earth that arises from its spinning motion on its axis, treating Earth as a uniform solid sphere.
J · s
(b) Calculate the angular momentum of Earth that arises from its orbital motion about the Sun, treating Earth as a point particle.
J · s
To calculate the angular momentum of Earth arising from its spinning motion on its axis, we need to know the relevant quantities: the mass of Earth (M), its radius (R), and its rotational angular velocity (ω).
(a) The formula for the angular momentum of a rotating object is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the rotational angular velocity.
For a uniform solid sphere, the moment of inertia (I) can be calculated as I = (2/5)MR^2.
So, the angular momentum (L) from Earth's spinning motion can be calculated as:
L = Iω
L = (2/5)MR^2 ω
Now, let's calculate it! We need to find the values of M, R, and ω.
- The mass of Earth (M) is approximately 5.97 × 10^24 kg.
- The radius of Earth (R) is approximately 6,371 km, which is equivalent to 6.371 × 10^6 meters.
- The rotational angular velocity (ω) can be expressed as the number of rotations per unit of time. Earth completes one rotation in approximately 24 hours, which equals 86,400 seconds.
Now, we can plug in the values and calculate the angular momentum (L):
L = (2/5)(5.97 × 10^24 kg)(6.371 × 10^6 m)^2 (2π radians)/(86,400 s)
The result will be in Joule-seconds (J·s).
(b) To calculate the angular momentum of Earth arising from its orbital motion about the Sun, we need to know the relevant quantities: the mass of Earth (M), its orbital radius (r), and its orbital angular velocity (ω).
The formula for the angular momentum of a point particle in circular motion is L = mvr, where L is the angular momentum, m is the mass, v is the orbital velocity, and r is the orbital radius.
For Earth, we need to consider it as a point particle moving in a circular orbit around the Sun.
- The mass of Earth (M) is the same as mentioned above: approximately 5.97 × 10^24 kg.
- The orbital radius (r) of Earth's orbit around the Sun is approximately 150 million kilometers, which is equivalent to 1.5 × 10^11 meters.
- The orbital angular velocity (ω) can be calculated using the formula ω = v/r, where v is the orbital velocity. Earth's orbital velocity can be calculated using Kepler's law: v = √(GM/r), where G is the gravitational constant (approximately 6.67430 × 10^-11 N·m^2/kg^2).
Now we can calculate the angular momentum (L):
L = mvr
L = (5.97 × 10^24 kg)(√(GM/r))(r)
The mass (m) cancels out in this equation because it is the same as the mass of Earth (M). We need to calculate the velocity (v) using Kepler's law and then substitute it into the equation.
L = (5.97 × 10^24 kg)(√(G(5.97 × 10^24 kg)/r))(r)
The result will be in Joule-seconds (J·s).