Motion in a Rotating Reference Frame

In summary, when breaking up the equation relating the accelerations ar and af into x and y components, we must consider the Coriolis, centrifugal, and Euler accelerations in addition to the acceleration due to rotation of the frame.
  • #1
garibaldi729
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Homework Statement



A particle moves in a rotating reference frame along the x-axis as x(t) = xo eat (xo and a are positive constants). The frame rotates with a time-dependant angular frequency ω(t) about the x-axis. The true physical force is in the x-direction of the rotating frame. Break up the equation relating the accelerations ar and af into x and y components.

Homework Equations



Equation relating frames' accelerations:
af = Af + ar + ω' x r + 2ω x vr + ω x (ω x r)

The Attempt at a Solution



I know that Af is zero because the frame's origin is not moving. I am just stuck when it comes to considering what is with respect to what. In my mind everything that is relative to the rotating frame (everything with an "r" subscript) has only an x component because motion is only happening in the x direction. And af is due to the true physical force which is also acting in the x direction. So r would be the only thing with a y component?

I am getting so mixed up with these inertial/non-inertial reference frames! What is going on here?
 
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  • #2


Hello! As a fellow scientist, I can understand your confusion when it comes to inertial and non-inertial reference frames. Let's break down the equation and see if we can make sense of it.

First, let's define the components of the equation:

- af: the acceleration in the fixed (inertial) frame
- Af: the acceleration in the rotating (non-inertial) frame
- ar: the acceleration due to the rotation of the frame
- ω: the angular velocity of the frame
- ω': the time derivative of ω (angular acceleration)
- r: the position vector of the particle

Now, as you correctly pointed out, Af is zero because the origin of the rotating frame is not moving. So let's focus on the other terms in the equation.

ar is the acceleration due to the rotation of the frame. This means that it is perpendicular to the x-axis (since the frame is rotating about the x-axis). So, ar only has a y component.

Next, we have ω' x r, which represents the Coriolis acceleration. This term is also perpendicular to the x-axis, so it only has a y component.

The term 2ω x vr represents the centrifugal acceleration. This term is parallel to the x-axis, so it only has an x component.

Finally, we have ω x (ω x r), which represents the Euler acceleration. This term is perpendicular to both the x-axis and the y-axis, so it has both x and y components.

Putting it all together, we can write the equation for the x and y components of the total acceleration (af + ar):

ax = 2ωyv + ω^2x

ay = -ar + ω'x - 2ωxv + ω^2y

I hope this helps clarify things for you. Remember, when dealing with non-inertial reference frames, it's important to consider all the different types of accelerations that can occur. Keep practicing and you'll get the hang of it!
 

What is a rotating reference frame?

A rotating reference frame is a coordinate system in which the observer is rotating along with the objects being observed. This means that the observer's point of view is constantly changing and the laws of physics may appear different in this frame compared to a stationary reference frame.

What is the Coriolis effect?

The Coriolis effect is a phenomenon that occurs in a rotating reference frame. It is the apparent deflection of a moving object due to the rotation of the reference frame. This effect is responsible for the rotation of hurricanes and the direction of wind patterns on Earth.

How does motion in a rotating reference frame affect objects?

Motion in a rotating reference frame can affect objects in several ways. The Coriolis effect can cause objects to appear to curve or deviate from their expected path. Objects may also experience centrifugal and centripetal forces, which can affect their speed and direction of motion.

What are some examples of motion in a rotating reference frame?

Some examples of motion in a rotating reference frame include the rotation of the Earth and other planets, the motion of objects on a spinning carousel, and the flight of airplanes and satellites in the Earth's rotating atmosphere.

How is motion in a rotating reference frame related to Newton's laws of motion?

Motion in a rotating reference frame is related to Newton's laws of motion because these laws still apply in a rotating frame, but may appear differently due to the effects of rotation. The first law of motion, also known as the law of inertia, still holds true in a rotating reference frame. The second law, which relates force and acceleration, can be modified to include the effects of the Coriolis force. And the third law, which states that for every action there is an equal and opposite reaction, also applies in a rotating frame.

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