Angular Velocity of Earth

In summary, the duration of one day on Earth would be 18 hours if the Earth were to suddenly shrink to 3/4 of its initial radius and 2/3 of its initial mass, assuming that angular momentum is conserved. This is calculated by first finding the new moment of inertia using the equation I = (2/3)I_original, and then finding the new angular velocity using the equation Iω = I_originalω_original.
  • #1
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Homework Statement


Find the hours it takes for the Earth to rotate in one day if the mass is reduced to 2/3 of its original mass and it is shrunk to 3/4 its original size.

Homework Equations



L=Iω

The Attempt at a Solution



To start this problem I assumed that angular momentum is conserved from the original Earth to the final earth. This means the angular velocity must change in order for the angular momentum to remain unchanged. So used this equation: Iearthωearth=Ikωk and got that Iearthωearth/Ikk. I plugged in the data and got that the new speed is twice the speed of the original earth. This would mean the Earth rotates about 12 hrs a day. This SEEMS wrong but I am not sure.

Note: k is the final earth.
 
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  • #2
Um.
I would be inclined to do this in two steps. First assume that the mass changed as a result of density, for a start - giving an identical 'size' of sphere as it started off with. That gives a new I. (2/3 of the original) But I can't see how this is relevant to the final answer, actually, because you could have two, spheres rotating around a common axis and each sphere could reduce in size, giving the same answer.
Then conserve angular momentum for the new reduced radius to find the new angular velocity. I is proportional to radius squared so new I is 9/16 of original. this gives a day length of 24X9/16 = 18hours

Buit the question is SLOPPY because what does "size" mean? Volume or radius?
 
  • #3
OKay, well I didn't copy the exact questions...Here: Suppose the Earth were to suddenly shrink to 3/4 of its initial radius and 2/3 of its initial mass. What would the duration of one day be?
 
  • #4
Haha. Sloppy student not sloppy question.
I guess the 9/16 is what you're after then.
 
  • #5


Your approach and calculations are correct. It may seem counterintuitive that reducing the mass and size of the Earth would result in a faster rotation, but this is due to the conservation of angular momentum. As the Earth's size and mass decrease, its moment of inertia (I) also decreases, causing the angular velocity (ω) to increase in order to maintain a constant angular momentum (L=Iω). Therefore, the Earth would rotate in approximately 12 hours instead of the 24 hours it currently takes. This is a very interesting concept and highlights the interconnectedness of different physical quantities in our universe. Great job on your solution!
 

What is Angular Velocity?

Angular velocity is a measure of how fast an object is rotating or spinning around a fixed point. It is usually measured in radians per second (rad/s) or degrees per second (deg/s).

What is the Angular Velocity of Earth?

The angular velocity of Earth is approximately 0.00007292 radians per second (rad/s) or 0.0041667 degrees per second (deg/s). This means that Earth completes one full rotation in about 23 hours, 56 minutes, and 4 seconds.

How is Angular Velocity of Earth calculated?

The angular velocity of Earth can be calculated by dividing the Earth's circumference (the distance around its equator) by the length of a day (the time it takes for Earth to complete one rotation). This calculation results in the Earth's angular velocity in radians per second or degrees per second.

What factors affect the Angular Velocity of Earth?

The angular velocity of Earth is affected by factors such as its mass, distance from the sun, and gravitational pull from other celestial bodies. It can also be affected by external forces, such as meteor impacts or volcanic eruptions, but these changes are very small and have little impact on Earth's overall angular velocity.

Why is the Angular Velocity of Earth important?

The angular velocity of Earth is important because it determines the length of a day and affects the Earth's rotation, which in turn affects our daily lives. It also plays a role in phenomena such as ocean tides and the Coriolis effect, which influence weather patterns and ocean currents. Understanding the Earth's angular velocity is crucial for studying and predicting these natural processes.

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