Calculate the kinetic energy that the earth has because of its rotation about its own axis. For comparison, the total energy used in the United States in one year is about 9.33 109 J.
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(b) Calculate the kinetic energy that the earth has because of its motion around the sun.
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To calculate the kinetic energy of the Earth due to its rotation about its own axis, we need to know its rotational inertia and angular velocity.
The rotational inertia (I) of the Earth can be approximated as a solid sphere. The formula for the rotational inertia of a solid sphere is given by:
I = (2/5) * m * r^2
where m is the mass of the Earth and r is its radius.
The angular velocity (ω) of the Earth can be calculated using the formula:
ω = 2π / T
where T is the period of the Earth's rotation, which is approximately 24 hours or 86,400 seconds.
Once we have obtained the values for I and ω, we can use the formula for rotational kinetic energy:
KE = (1/2) * I * ω^2
To calculate the kinetic energy of the Earth due to its motion around the Sun, we need to know its mass (m), orbital radius (r), and orbital velocity (v).
The mass of the Earth can be obtained from scientific sources, and the orbital radius is approximately 149.6 million kilometers. The orbital velocity can be calculated using the formula:
v = 2π * r / T'
where T' is the period of Earth's orbit around the Sun, which is approximately 365.25 days or 31,557,600 seconds.
Once we have obtained the values for m, r, and v, we can use the formula for translational kinetic energy:
KE = (1/2) * m * v^2
By plugging in the appropriate values into these formulas, we can calculate the kinetic energy of the Earth due to its rotation and orbital motion.