Question

You found the two main angular velocities of the Earth: one due to the Earth's motion around the sun, and one due to its rotation about its own axis. Now let's figure out the energy and momentum associated with that motion. For the purposes of this problem, treat the Earth as a solid, uniform sphere with mass 5.97×10²⁴ kg and radius 6.37×10⁶ m , and assume that the Earth's orbit around the sun is circular with a radius of 1.5×10¹¹ m .

Part A) What is the angular kinetic energy of the Earth due to its orbit around the sun?

Part B) What is the magnitude of the Earth's angular momentum due to its orbit around the sun?

Part C) What is Earth's angular kinetic energy due to its rotation around its axis?

Part D) What is the magnitude of the Earth's angular momentum due to its rotation around its axis?

Part E) Which of the following best explains where the Earth's angular kinetic energy and momentum came from?

Remember that energy and momentum are always conserved (though energy can change forms). In other words, if you start with a certain amount of energy and momentum, you must end with the same amount of energy and momentum. By conservation of energy and momentum, the values you've calculated in this problem must have come from somewhere.

Which of the following best explains where the Earth's angular kinetic energy and momentum came from?

a. The solar system formed from a massive cloud of gas and dust, which was slowly rotating. As the cloud collapsed under its own gravitational pull, the cloud started to spin faster, just as an ice skater pulling his arms in will spin faster. Because all of the material that accreted to form the planet was rotating, the planet was rotating as well.

b. As the Earth formed, it experienced a series of collisions with asteroids and comets. These asteroids and comets hit the ball of rock that was forming into the planet off-center. Over time, the off-center collisions gradually caused the planet to rotate faster.

c. As the Moon orbits around the Earth, it creates tides on the Earth. Over time the tides have caused the Earth to rotate faster and faster.

d. Sheer force of will.

Answer 1

Here we will use equations of rotational kinetic energy and angular momentum

K.E = 1/2 * I * w²

L = Iw

where I is moment of inertia and w is angular velocity

----------------------------------------------------------

(a) rotational kinetic energy is given as

K.E = 1/2 * I * w²

where I = m * r² =  5.97e24 * 1.5e11² = 1.343e47 Kg.m²

w = 2 * pi / T

where T is time taken to complete one orbit = 365 days = 3.153e7 seconds

w = 1.992e-7 rad/sec

so,

K.E = 2.665e33 J

--------------------------------------------------

(b) angular momentum

L = Iw

L = 1.343e47 * 1.992e-7

L = 2.675e40 Kg.m² / s

----------------------------------------------

(c) K.E ( about rotation axis)

rotational kinetic energy is given as

K.E = 1/2 * I * w²

where I = 2/5 * m * r² = 2/5 * 5.97e24 * 6.37e6² = 9.689e37 Kg.m²

w = 2 * pi / T

where T is time taken to complete one rotation = 24 hours = 86400 seconds

w = 7.27e-5 rad/sec

so,

K.E = 2.56e29 J

----------------------------------------------

(d) angular momentum about rotation axis

L = 9.689e37 * 7.27e-5

L = 7e33 Kg.m² /s

------------------------------------------------

(e)

option (a) seems correct as momentum seems to be conserved here