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 axis and its motion around the Sun, we need to consider two separate components: rotational kinetic energy and orbital kinetic energy.
(a) Rotational Kinetic Energy:
The Earth's rotational kinetic energy can be calculated using the formula:
KE_rot = (1/2) * I * ω^2
where:
KE_rot is the rotational kinetic energy,
I is the moment of inertia of the Earth about its axis of rotation,
and ω is the angular velocity of the Earth.
To calculate the moment of inertia (I) for a uniform sphere, we can use the formula:
I = (2/5) * m * r^2
where:
m is the mass of the Earth,
and r is the radius of the Earth.
The angular velocity (ω) is given by:
ω = 2π / T
where T is the time taken for one complete rotation. For Earth, T is approximately 24 hours or 86,400 seconds.
(b) Orbital Kinetic Energy:
The Earth's orbital kinetic energy can be calculated using the formula:
KE_orb = (1/2) * m * v^2
where:
KE_orb is the orbital kinetic energy,
m is the mass of the Earth,
and v is the orbital velocity of the Earth around the Sun.
To calculate the orbital velocity (v), we can use the formula:
v = 2πr / T_orbit
where r is the average distance between the Earth and the Sun (approximately 149.6 million km or 9.3 million miles), and T_orbit is the time period of one complete orbit around the Sun (approximately 365.25 days or 31,557,600 seconds).
Once we have the rotational and orbital kinetic energies, we can add them together to obtain the total kinetic energy of the Earth.
It is important to note that the given comparison value of the total energy used in the United States is used for context and is not directly related to the calculation of the Earth's kinetic energy.