(a) Calculate the kinetic energy that the earth has because of its rotation about its own axis. 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 9.33 109 J.
(b) Calculate the kinetic energy that the earth has because of its motion around the sun.
Please check. The answer i came up with for a. is wrong.
a.
rotational KE = 1/2(Iw^2)
=1/2(2/5mr^2)w^2
=1/2((2/5(5.98x10^24kg)(6.38x10^6m))(2PI rad/ s)^2
=(7.63048x10^30kg*m)(39.4384/2.49x10^-13)
= 1.210 x 10^18
ur answer is right. I do belive
I don't understand where you got the w^2 The earth's rotational w is
(2 pi rad)/(24*3600 sec) = 7.27*10^-5 rad/s
To calculate the kinetic energy (KE) of the Earth due to its rotation about its own axis, we can use the formula:
KE = 1/2 * I * w^2
Where:
- KE is the kinetic energy
- I is the moment of inertia of the Earth (assuming a uniform sphere)
- w is the angular velocity of the Earth's rotation
The moment of inertia of a uniform sphere is given by the formula:
I = (2/5) * m * r^2
Where:
- m is the mass of the Earth
- r is the radius of the Earth
Next, we need to find the mass and radius of the Earth:
- Mass of the Earth (m) = 5.98 x 10^24 kg
- Radius of the Earth (r) = 6.38 x 10^6 m
Now, let's calculate:
I = (2/5) * (5.98 x 10^24 kg) * (6.38 x 10^6 m)^2
= 9.46 x 10^37 kg m^2
Given that the angular velocity (w) of the Earth's rotation is 7.27 x 10^-5 rad/s, we can substitute these values into the KE formula:
KE = 1/2 * (9.46 x 10^37 kg m^2) * (7.27 x 10^-5 rad/s)^2
= 1.21 x 10^31 J
So, the kinetic energy that the Earth has due to its rotation about its own axis is approximately 1.21 x 10^31 J.
For part b, to calculate the kinetic energy of the Earth due to its motion around the sun, we can use the formula:
KE = 1/2 * m * v^2
Where:
- KE is the kinetic energy
- m is the mass of the Earth
- v is the velocity of the Earth's orbit around the sun
The velocity (v) can be found using the formula for the orbit circumference:
C = 2 * π * r
Where:
- C is the circumference of the Earth's orbit
- r is the average distance from the Earth to the sun
The average distance from the Earth to the sun is approximately 150 million km or 1.5 x 10^11 m.
C = 2 * π * (1.5 x 10^11 m)
= 9.42 x 10^11 m
The time it takes for the Earth to complete one orbit around the sun is approximately 365.25 days or 3.15576 x 10^7 seconds.
Now, we can calculate the velocity:
v = C / t
= (9.42 x 10^11 m) / (3.15576 x 10^7 s)
= 2.98 x 10^4 m/s
Finally, substitute the mass of the Earth (5.98 x 10^24 kg) and the velocity (2.98 x 10^4 m/s) into the KE formula:
KE = 1/2 * (5.98 x 10^24 kg) * (2.98 x 10^4 m/s)^2
= 2.68 x 10^29 J
Therefore, the kinetic energy that the Earth has due to its motion around the sun is approximately 2.68 x 10^29 J.