Inertial/non-inertial reference frames

In summary, the conversation discusses a question about finding the components of angular velocity in a reference frame fixed to the surface of a rotating Earth. It also involves solving an ODE and integrating to find a general solution, as well as using series expansion to find a particular solution. The final part of the conversation discusses how an inertial observer would account for the eastward deflection of a falling particle in this scenario.
  • #1
Rhi
10
0
I'm a bit unsure about the last couple of bits of this question, and I'm hoping someone might be able to help.

Homework Statement



a) Let a reference frame with origin O & Cartesian axes (x, y, z) be fixed relative to the surface of the rotating Earth at co-latitude θ (i.e. 0≤θ≤∏, where θ = 0 corresponds to the north pole). Increasing x is east, increasing y is north & increasing z is upwards (opposite direction to gravity g). The Earth is assumed to rotate steadily with angular velocity ω. Find the components of ω in this frame of reference. Ignoring the centrifugal force, show that the motion of a particle of mass m under gravity is governed by

[itex]\ddot{x} − 2ω\dot{y} cos θ + 2ω\dot{z} sin θ = 0 [/itex]
[itex]\ddot{y}+ 2ω\dot{x} cos θ = 0 [/itex]
[itex]\ddot{z}− 2ω\dot{x} sin θ = −g [/itex]

where ω = |ω| and g = |g|. Assuming θ is constant, by integrating the second and third of these equations with respect to time and substituting into the first equation, show that

[itex]\ddot{x}+ 4ω^{2}x= 2ω(v_{0}cosθ-w_{0}sin)+2gtsinθ [/itex]

where v0 and w0 are constants. Hence find the general solution for x.

b) If a particle falls from rest at O, find x as a function of t. The particle falls only for a brief time before it hits the ground, so that ωt is small throughout its motion. Use a series xpansion of solution for x in ωt to show, to leading order,

[itex]x =\frac{1}{3}gωt^{3} sin θ [/itex]

c) Explain briefly how an inertial observer would account for this eastward deflection
of the falling particle.

Homework Equations



[itex]m\textbf{a}=\textbf{F}-m\dot{\textbf{ω}}\times\textbf{r}-2m\textbf{ω}\times\dot{\textbf{r}}-m\textbf{ω}\times(\textbf{ω}\times\textbf{r})-m\textbf{A} [/itex]


The Attempt at a Solution



For a) I get ω=ωsinθy+ωcosθz, and using the equation above, with the fact that ω is constant and ignoring the centrifugal force, I get the three equations as stated. Integrating then gives [itex]\ddot{x}+ 4ω^{2}x= 2ω(v_{0}cosθ-w_{0}sin)+2gtsinθ [/itex].

solving this as a 2nd order ODE, I get complementary solution [itex]x=αcos(2ωt)+βsin(2ωt) [/itex] and particular solution [itex]x=\frac{gsinθ}{2ω}t+\frac{v_{0}cosθ-w_{0}sinθ}{2w^{2}} [/itex]

so the general solution for x is these added together.

For b), I plugged in the initial values, at t=0, x=0, [itex]\dot{x}[/itex]=0 to get

[itex]α=\frac{-(v_{0}cosθ-w_{0})}{(2ω^{2})}[/itex]

[itex]β=\frac{-gsinθ}{4ω^{2}}[/itex]

and using the series expansions for sin and cos, I get for small t

[itex]x=\frac{2ω^{2}t^{2}(v_{0}cosθ-w_{0}sinθ)+2/3gsinθωt^{3}}{2ω^{2}}[/itex]

which simplifies to [itex]x=(v_{0}cosθ-w_{0}sinθ)t^{2}+1/3gsinθt^{3}[/itex]

The answer I'm supposed to get here is just the second term, but I'm not entirely sure if I've done this right. Can I just cancel the first term here as t is small?

I'm also not entirely sure what answer part c) is looking for. Is it anything to do with Coriolis?

I'd be grateful if anyone could shed a bit of light on this. Thanks!
 
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  • #2
[itex]v_{0}[/itex] and [itex]w_{0}[/itex] are constants of integration. What would they be for the initial conditions given in b)?
 
  • #3
Apologies if I'm being really stupid here, but I don't see how the initial conditions give any information about v_{0} and w_{0}? The two initial conditions give you the constants from the ODE don't they?
 
  • #4
You got them by integrating the equations for y and z. b), however, specifies that the particle is initially at O at rest.
 
  • #5
Ah right of course! I see what you mean now, and that gives the correct answer, thanks.

The only other bit I wasn't sure about was part c), I don't really know what sort of answer they're looking for..
 
  • #6
I think you should take an inertial frame of reference whose origin is coincident with O at t = 0 and see what the motion would look like in that frame (when t is small). I think in should boil down to the fact that in an inertial frame the motion should be the familiar parabolic motion in the XZ plane.
 
  • #7
Hmm.. How do you get parabolic motion? sorry if I'm being slow here!
 
  • #8
Imagine an inertial frame coincident with the rotating frame but fixed with respect to the center of the planet. If you release at particle at rest in the rotating frame, in the inertial frame it will have a non-zero horizontal velocity. The only force in the inertial frame acting on it will be gravity. The resultant motion is parabolic.
 
  • #9
Oh, right, I see what you mean now... I was thinking that it was released from rest so would have zero velocity in the x direction so wouldn't be parabolic but that's not right if you're in an inertial frame.. Thanks for all the help!
 

1. What is the difference between an inertial and non-inertial reference frame?

An inertial reference frame is a frame of reference in which Newton's first law of motion holds true, meaning that an object at rest will remain at rest and an object in motion will continue in a straight line at a constant speed unless acted upon by an external force. In contrast, a non-inertial reference frame is one in which this law does not hold true, typically due to the presence of non-zero accelerations or forces.

2. How do you determine if a reference frame is inertial or non-inertial?

To determine if a reference frame is inertial or non-inertial, one must consider whether or not the frame is accelerating. If the frame is accelerating, it is non-inertial. If the frame is not accelerating, it is inertial.

3. Can a non-inertial reference frame be used to make accurate measurements?

Yes, a non-inertial reference frame can be used to make accurate measurements, but only if the effects of the non-inertial forces or accelerations are taken into account. For example, in a rotating reference frame, the Coriolis force must be considered in order to make accurate measurements.

4. What is the principle of relativity in reference frames?

The principle of relativity states that the laws of physics are the same in all inertial reference frames. This means that the fundamental laws of motion, such as Newton's laws, will hold true regardless of the reference frame used to observe them.

5. How are inertial and non-inertial reference frames used in real-world applications?

Inertial and non-inertial reference frames are used in various fields, such as physics, engineering, and aerospace, to understand and analyze the motion of objects. They are also used in navigation systems, such as GPS, to accurately determine the position and velocity of objects. In addition, understanding reference frames is crucial for accurately predicting and planning the motion of objects in space and other non-inertial environments.

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