Calculate the kinetic energy that the earth has because of (a) its rotation about its own axis and (b) its motion around the sun. Assume that the earth is a uniform sphere and that its path around the sun is circular. For comparison, the total energy used in the United States in one year is about 1.1 x 1020 J.
To calculate the kinetic energy of the Earth due to its rotation about its own axis and its motion around the Sun, we need to use the formulas for rotational and circular motion.
(a) Kinetic Energy due to Earth's rotation:
The formula for the rotational kinetic energy of a uniform sphere is given by:
KE_rot = (2/5) * I * ω²
Where KE_rot is the rotational kinetic energy, I is the moment of inertia of the sphere, and ω is the angular velocity of the rotation.
The moment of inertia of a uniform sphere is given by:
I = (2/5) * m * R²
Where m is the mass of the sphere and R is its radius.
To find the angular velocity ω, we can use the period of Earth's rotation, T_rot, which is approximately 24 hours or 86400 seconds. The angular velocity ω is given by:
ω = (2π) / T_rot
Substituting these values into the formulas, we get:
KE_rot = (2/5) * [(2/5) * m * R²] * [(2π / T_rot)²]
(b) Kinetic Energy due to Earth's motion around the Sun:
The formula for the kinetic energy of an object in circular motion is given by:
KE_circ = (1/2) * m * v²
Where KE_circ is the kinetic energy, m is the mass of the object, and v is its velocity.
The velocity of the Earth in its circular orbit around the Sun, v, is given by:
v = (2πR_sun) / T_orbit
Where R_sun is the radius of Earth's orbit around the Sun, and T_orbit is the period of Earth's orbit, which is approximately 365.25 days or 3.154 x 10^7 seconds.
Substituting these values into the formula, we get:
KE_circ = (1/2) * m * [(2πR_sun / T_orbit)²]
To calculate the total kinetic energy of the Earth, we simply add the kinetic energies from rotation and circular motion:
Total KE = KE_rot + KE_circ
Now, let's plug in the values to calculate the kinetic energies:
Given:
Mass of the Earth, m = 5.97 x 10^24 kg
Radius of the Earth, R = 6.37 x 10^6 m
Radius of the Earth's orbit, R_sun = 1.50 x 10^11 m
Calculations:
T_rot = 24 hours = 24 x 3600 seconds = 8.64 x 10^4 seconds
T_orbit = 365.25 days = 365.25 x 24 x 3600 seconds = 3.154 x 10^7 seconds
ω = (2π) / T_rot = (2π) / (8.64 x 10^4) = 7.27 x 10^(-5) rad/s
KE_rot = (2/5) * [(2/5) * (5.97 x 10^24) * (6.37 x 10^6)²] * (7.27 x 10^(-5))²
v = (2πR_sun) / T_orbit = (2π x 1.50 x 10^11) / (3.154 x 10^7) = 2.98 x 10^4 m/s
KE_circ = (1/2) * (5.97 x 10^24) * (2.98 x 10^4)²
Total KE = KE_rot + KE_circ
Now you can use a calculator to get the numerical values for the rotational and circular kinetic energies, as well as the total kinetic energy.