Derive the equation of effective force on a rotating frame about Earth

In summary, the textbook provides an expression for the effective force of a particle on a rotating frame, and an arbitrary vector Q can be used to calculate the Rdotdot w.r.t. the fixed frame. However, when deriving (10.31) from (10.12), the Rdotdot w.r.t. the rotating frame disappears. This is because the frame is rotating around the Earth and the time derivative of R with respect to the rotating frame remains unchanged. It is the object's distance from the frame, denoted as r, that is changing.
  • #1
Tony Hau
101
30
In my textbook, the effective force of a particle on a rotating frame is given as below:
1586681268592.png

The diagram is:
1586681310177.png

What I do not understand is the expression for Rf dotdot, which is given as below:
1586681436953.png


According to the book, an arbitary vector Q can be expressed as:
1586681521746.png

So Rdotdot w.r.t fixed frame can be obtained by substituting Q for Rdot

Why the Rdotdot w.r.t. the rotating frame is gone when we derive (10.31) from (10.12)? As the frame is moving away from the centre of Earth, the R, which is the distance between the frame and the centre of the Earth, should be changing as well when measured from the moving frame.
 
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  • #2
I just figure out the answer. I think it is because the frame is not moving with respect to the centre of the Earth. It is the object, whose distance from the moving frame is denoted as small letter r, that is changing. So the time derivative of R with respect to the rotating frame is not changing. The frame is only rotating around the Earth and so the time derivative with respect to the fixed Earth frame is omega cross r.
 
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Related to Derive the equation of effective force on a rotating frame about Earth

1. What is the equation for effective force on a rotating frame about Earth?

The equation for effective force on a rotating frame about Earth is F = ma + 2mω x v + mω^2 x r, where F is the effective force, m is the mass of the object, a is the acceleration, ω is the angular velocity of the rotating frame, v is the velocity of the object, and r is the distance from the center of rotation.

2. How is the effective force on a rotating frame about Earth different from the force on a stationary frame?

The effective force on a rotating frame about Earth includes an additional term, 2mω x v, which accounts for the Coriolis force. This force arises due to the rotation of the frame and can cause objects to experience a deflection in their motion.

3. Can the equation for effective force on a rotating frame about Earth be used for any object?

Yes, the equation for effective force on a rotating frame about Earth can be used for any object, as long as its motion is being observed from a rotating frame of reference. This equation takes into account the effects of both the Coriolis force and the centrifugal force.

4. How does the direction of the effective force change as the object moves in a rotating frame about Earth?

The direction of the effective force changes as the object moves in a rotating frame about Earth due to the presence of the Coriolis force. This force acts perpendicular to both the velocity of the object and the axis of rotation, causing a deflection in the object's motion.

5. Are there any simplifications or assumptions made in the equation for effective force on a rotating frame about Earth?

Yes, the equation for effective force on a rotating frame about Earth assumes that the frame of reference is rotating at a constant angular velocity and that the object is moving at a constant velocity. It also assumes that the object is not experiencing any other external forces besides the effective force.

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