Jumping up on a rotating Earth

In summary: When a force acts in a direction that is not at right angles to the radius then no work is done. Work done only depends on the force component in the direction of motion. In summary, the linear momentum of the man jumping on the equator will remain constant, while the angular momentum will also remain constant due to the vector cross product. The angle between the radius and the velocity vector increases, but the cross product stays the same over time. This means that the angular velocity with respect to the center of rotation must decrease.
  • #1
Guywithquestions
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Hello,

I know it suppose to be a relatively basic question but still somehow I can't fully understand it.
Let assume that a man jumps vertically on the equator, while the Earth is of course rotating. What will happen to the value of his linear momentum in the horizontal axis?
It seems to me that if the angular momentum must be conserved, than the linear momentum must decrease, because:
J = p x r
And since r is increased due to the jump
p must decrease
Is it correct? If so how is it possible since both angular momentum and linear momentum must be conserved?

Thank you kindly.
 
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  • #2
The angular and linear momentum of the system (earth+man) will be conserved (because there are by assumption ~no external forces or torques on the system). The man, depending upon how he jumps, can change momentum for himself (and exactly equally and oppositely for the earth).
 
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  • #3
Guywithquestions said:
Summary: If a man jumps off the equator, What will happen to the value of his linear momentum in the horizontal axis?

Hello,

I know it suppose to be a relatively basic question but still somehow I can't fully understand it.
Let assume that a man jumps vertically on the equator, while the Earth is of course rotating. What will happen to the value of his linear momentum in the horizontal axis?
It seems to me that if the angular momentum must be conserved, than the linear momentum must decrease, because:
J = p x r
And since r is increased due to the jump
p must decrease
Is it correct? If so how is it possible since both angular momentum and linear momentum must be conserved?

Thank you kindly.
His linear momentum remains constant. So while the radius for calculating angular momentum increases his angular velocity with respect to the center of rotation must decrease.

Below is a quick vector diagram of the situation:
The green arrow is his Vertical jump speed
Blue arrow is his tangential velocity due to the rotation of the Earth
Red arrows are his resultant velocity broken into two sections of equal time.
Cyan lines represent the angles swept out by the radial line.
Note that for the first time section of his trajectory the cyan line has swept through a larger angle than it does during the second time section of equal length. The cyan line grows in length but rotates slower as time goes on.
Momentum.png
 
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  • #4
Thank you for answering!
My problem rised from the equation j=p x r
And since r is increased and p (linear momentum) stays constant it seems as though j (angular momentum) must increase as well (contrary to staying constant).
If I understand correctly from your drawing it can be seen that the angle between the radius and the velocity vector increases and therefore the cross product stays the same over time.
 
  • #5
Guywithquestions said:
And since r is increased and p (linear momentum) stays constant it seems as though j (angular momentum) must increase as well (contrary to staying constant).
Remember that angular momentum is not just the product of the magnitudes of r and p. It is the vector cross product. One way to understand the cross product is that it is not p that counts. It is only the component of p that is at right angles to r.
 

1. How does jumping up on a rotating Earth affect our weight?

Jumping up on a rotating Earth does not affect our weight. Our weight is determined by our mass and the gravitational pull of the Earth, which remains constant regardless of our position on the rotating Earth.

2. Will jumping up on a rotating Earth cause us to fly off into space?

No, jumping up on a rotating Earth will not cause us to fly off into space. The Earth's gravity is strong enough to keep us grounded, even as it rotates.

3. Why do we feel like we are moving when we jump up on a rotating Earth?

We feel like we are moving when we jump up on a rotating Earth because of inertia. Our bodies have a tendency to resist changes in motion, so when the Earth rotates, our bodies also rotate with it, giving us the sensation of movement.

4. Does the direction of the Earth's rotation affect how high we can jump?

No, the direction of the Earth's rotation does not affect how high we can jump. Our jumping ability is determined by our muscle strength and the force we apply, not by the direction of the Earth's rotation.

5. Can jumping up on a rotating Earth cause a change in the Earth's rotation speed?

No, jumping up on a rotating Earth does not have a significant impact on the Earth's rotation speed. The Earth is a massive object with a strong angular momentum, so the force of our jumps is negligible in comparison.

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