How to prove Coriolis acceleration

These two reference systems can rotate with each other. For example, when the high-speed centrifuge is started, the test tube reference frame and the desktop reference frame rotate relatively. The particles in the test tube move in a straight line along the test tube, but spiral relative to the desktop, so we also need to transform the rotating coordinate system.

Consider the disk S' rotating relative to the desktop S, as shown in Figure 2- 17. Let the rotational angular velocity ω be a constant vector, pointing to the positive direction perpendicular to the Z axis of the disk surface, with the rotational axis at the center O' of the disk surface and the origin o of the desktop coincident with it. Suppose that vector a is fixed on S'. Note that the velocity expression (2.2. 10) shows that the increment of a in dt time is

dA=A(t+ dt)- A(t)=(ω×A)dt

If the vector has an increment dA' relative to s' at the same time, the increment relative to s is

Da = (ω× a) dt+da So we have a general relationship:

Or write a symbolic equation:

Obviously, the transformation relation of speed can be obtained by substituting the position vector into the above formula:

The derivative with apostrophe in the formula only means that it is carried out in the S' system, and does not mean that there is any difference in time. This also applies to other vectors. For example, any vector can be replaced by two vectors starting from the origin. The above practice can be completely extended to three-dimensional situations. The symbolic equation (2.7.2) is linear (satisfying the distribution law). For the velocity vector, we have

As can be seen from the observers in the S system, the acceleration consists of three parts. The first item is in the S' system.

Acceleration. When the particle is stationary in the S' system, the meaning of the third term can be clearly seen:

ω×(ω×r)=-(ω ω)ρ (2.7.5)

That is, centripetal acceleration. The second term is called Coriolis acceleration, which has a non-zero value only when the particle moves in the S' system. Is the expression *(2.7.4) consistent with the expression of acceleration in plane polar coordinates (1.5)? If the angular velocity is not a constant vector, are equations (2.7.3) and (2.7.4) correct? If it is not correct, how should it be revised?

Let's discuss the influence of the earth's rotation. The rotating earth is considered as an S' system, and a "non-rotating" earth (translation coordinate system) is an S system. In the earth reference system, the acceleration of gravity of particles is

g = G0-2ω×v′-ω×(ω×r)(2 . 7 . 6)

we know

G0 ≈ 9.8m/s2

ω= 7.292× 10-5 radians/second

In contrast, the inertial centrifugal term is much smaller,

|ω×(ω×r)|≤ω2R≈3.39× 10-2m/S2 < < G0

Thus, if the effective acceleration of gravity is incorporated, the formula (2.7.6) can be written as follows.

mg = mg eff-2mω×v′(2 . 7 . 7)

The last term is the Coriolis force on a moving object. It should be noted that this term is completely transformed from the coordinate system, or it is produced because the observer's perspective in the rotating coordinate system is different from that in the translating coordinate system. Generally, we can say that Coriolis force is a kinematic effect. * Is Coriolis force related to latitude? Is there a difference between the southern hemisphere and the northern hemisphere?

According to formula (2.7.7), we can judge the deviation of falling body. Roughly speaking, the speed (zero order approximation) of a falling body is in the -r direction. For the northern hemisphere, we can judge that the speed will be biased to the east, that is, -2mω×v' ~ωk×er =ωEJ. The so-called falling body is eastward. If it is from (2.7)

* Discussion: Will the thrown object fall at the throwing point?

The movement of the earth's surface is also influenced by Coriolis force. As can be seen from Figure 2- 18, rotation causes the movement to deviate from the forward right hand direction. We can decompose the speed to obtain quantitative results:

-2ω×(vθeθ+vjej)= 2ω(vθeθ×k+vjej×k)

=2ω(-vθcosθej+vjeρ)

=2ωcosθ(-vθej+vjeθ)

+2ωvjsinθer

Because of the existence of the g term, the radial term in the formula can be ignored, and the first two terms accurately indicate that the acceleration points to the right hand side of the motion direction.

A typical example of Coriolis force is cyclone in the atmosphere. In the weather forecast program, you may have seen the counterclockwise cyclone in the satellite image. In the southern hemisphere, this cyclone is clockwise. Foucault's pendulum (18 19- 1868) is an excellent example to show the earth's rotation. 36866.88686868686

Coriolis force is also manifested in microscopic phenomena. For example, it complicates the vibration of rotating molecules and makes the energy spectrum of molecular rotation and vibration interact.

exercises

2. 1. A charged particle is injected with an electric field at a speed of 300× 106m/s, which is accelerated in the opposite direction by1.15×1014m/S2. How far do charged particles fly and then stay still? How long can it stand still?

2.2. In the past 1s, the free fall has passed half of its total distance. Try to find out the falling distance and the time required.

2.3. A steel ball began to fall freely from the roof of a building. An observer in the building stood in front of a window with a height of 1.3m, and found that it took 1/8s for the steel ball to fall from the top to the bottom of the window. The steel ball continued to fall. After 2.0s, it completely collided with the horizontal ground and rose to the bottom of the window, trying to find out the height of the building.

2.4. Design an experiment to measure the acceleration g of gravity by falling or throwing.

2.5. A ground radar observer is observing a projectile. At a certain moment, he got the following information: the projectile has reached the highest point, and the horizontal speed is V. At this time, the linear distance between the projectile and the observer is L, and the angle between the observer's line of sight and the horizontal plane is θ.

(a) Calculate the horizontal distance d between the impact point of the projectile and the observer, which is expressed by L, V and θ.

(b) Can the projectile fly over the observer's head?

2.6. Pulsars in the Crab Nebula, with a rotational speed of 30 r/s and a radius of 150 km. What is the acceleration of an object near the equator of a star?

2.7. In Bohr's model of hydrogen atom, the electron moves around the nucleus in a uniform circle, and the speed is 2.18 ×10-1m in an orbit with a radius of 5.29 × 106m/s, and it is calculated that the electron is in.

2.8. Because of the rotation of the earth, a certain point on the equator of the earth has a certain acceleration, and the earth itself has a certain acceleration because of its revolution around the sun. Try to find the ratio of these two accelerations. Suppose the orbit of revolution is circular.

2.9. A tuning fork and a standard tuning fork with a frequency of 384 Hz form a beat frequency of three times per second. When a small piece of wax sticks to one of the tuning forks, the beat frequency will decrease. What is the frequency of this tuning fork?

2. 10. Try to write a program to demonstrate Lissajous graphics. X = COS (2 π 1 T+0) can be used as an adjustable parameter with vertical vibration frequency and initial phase.

2. 1 1. Try to make a superposition diagram of vectors and displacements of two vibrations with orthogonal phase differences.

2. 12. The simple harmonic oscillator subjected to periodic impact is discussed. The impact caused a discontinuous jump in speed. What is its phase space trajectory? If we only take the points t=T, 2T, 3T, … in the phase space, what is the graph like?

2. 13. What is the phase diagram of the motion in the double-well potential?

2. 14. The pilot flies eastward from point A to point B and then returns. The speed of air relative to the ground is u, while the speed of aircraft relative to air is constant. Try to find out the round-trip flight time of wind speed: (a) zero, (b) due east and (c) due north. What are the requirements for V' in the latter two situations?

2. 15. A man wants to cross a river 500 meters wide. His speed relative to the water is 3000m/h, and the current speed is 2000m/h. Assuming that the walking speed of this person on the shore is 5000m/h, (a) What route does this person need to choose to reach the position facing the starting point on the other side of the river in the shortest time? (b) What is the shortest time?

2. 16. A virus particle with mass of m exists in the solution, and the test tube containing the solution is placed on the centrifuge. The centrifuge rotates at the speed of n revolutions per minute. At a certain moment, the virus is located at a certain distance from the rotation axis R, and moves radially with a constant speed v0 relative to the test tube. Try to consider the movement of the virus quantitatively, and give all the forces in the following two reference frames: (a) rotating with the centrifuge. (b) Fixed laboratory reference system.