Question about Formulae for Motion in a Rotating Reference Frame

In summary: Add these to the angular velocity and acceleration of the plate and you get the particle's angular velocity and acceleration in the non-rotating frame.In summary, the equations for velocity and acceleration in a rotating reference frame involve terms such as angular velocity, angular acceleration, and vectors from an origin point to various points on the object. The third term in the velocity equation and the second term in the acceleration equation are related to the particle's motion in the circular groove, while the other terms are related to the plate's rotation and the particle's motion relative to the center of the groove. It appears that the problem is using a non-rotating frame, and the given angular velocities and accelerations are relative to that frame.
  • #1
Master1022
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Homework Statement
Particle P moves in a circular groove with radius ## a ## which has been cut into a square plate with sides of length ## l ##. The plate rotates about its corner ## O ## with with angular velocity ## \omega \hat k ## and angular acceleration ## \dot \omega \hat k ##. Particle P rotates about the centre of the circular groove (which coincides with the centre of the square plate) with with angular velocity ## \alpha \hat k ## and angular acceleration ## \dot \alpha \hat k ##
Relevant Equations
## \ddot r = \ddot R + \ddot \rho + \omega \times \omega \times \rho + 2 \omega \times \rho ##
Hi,

I am reading the following question: "Particle P moves in a circular groove with radius ## a ## which has been cut into a square plate with sides of length ## l ##. The plate rotates about its corner ## O ## with with angular velocity ## \omega \hat k ## and angular acceleration ## \dot \omega \hat k ##. Particle P rotates about the centre of the circular groove (which coincides with the centre of the square plate) with with angular velocity ## \alpha \hat k ## and angular acceleration ## \dot \alpha \hat k ##"

I am just trying to understand what the terms in these equations for velocity and acceleration in a rotating reference frame mean. ## \vec R ## is defined to be the vector from some origin ## O ## to some point ## o ##, ## \vec \omega ## is defined to be the angular velocity of ## o ## about O, and ## \rho ## is defined to to be a vector from ## o ## to some point P. ## \vec r ## is defined to point from O to P.

My attempt:
First to start with the velocity equation (generic formula, terms aren't related to problem)
## \dot r_p = \dot R + \dot \rho + (\omega \times \rho) ##

This is what I think the terms are:
- ## \dot R ##: this will include the ## \omega_{O} \times r_{O/A} = \omega \hat k \times \frac{l \sqrt 2}{2} \hat j = \frac{- \omega l \sqrt 2}{2} \hat i ## term
- ## \dot \rho ##: this will include the ## \omega_{P/o} \times r_{P/o} = \alpha \hat k \times a \hat i = \alpha a \hat j ##
- ## \omega \times \rho ## term which will equal ## \omega_{O} \times r_{P/o} = \omega \hat k \times a \hat i = \omega a \hat j ##

I can understand what the first two terms are, but what is the third term? Originally I thought that ## \dot \rho ## would correspond to changes in the radius (which is fixed in this case and thus would be 0) and the ## \omega \times \rho ## would be what I have denoted ## \dot \rho ## as.

Now to consider the acceleration equation (once again the symbols are just generic and bear no relation to variable names in the problem)
## \ddot r_p = \ddot R + \ddot \rho + (\omega \times \omega \times \rho) + 2 \omega \times \dot \rho + (\dot \omega \times \rho) ##

- Coriolis term: ## 2 \omega \times \dot \rho = 2 \omega_{O} \times \dot \rho ## where ## \dot \rho ## was found above
- The ## (\omega \times \omega \times \rho) ## term. At first I thought this was centripetal of P relative to o, but the answer seems to denote it as a centripetal term relative to O: ## \omega_{O} \times \omega_{O} \times a ## (have dropped the vector directions for simplicity).
- The ## \ddot \rho ## term which includes the centripetal acceleration of P about o ## \alpha ^ 2 a ## and the tangential acceleration of P about o ## a \dot \alpha ##.
- ## \ddot R ##: which will include the centripetal acceleration of o around O and the tangential term ## \omega_{O} ^ 2 \frac{l \sqrt 2}{2} ##
- ## \dot \omega \times \rho = \dot \omega_{O} \times a ##

Are those correct interpretations of those terms?

Any help would be greatly appreciated
 
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  • #2
Master1022 said:
I am just trying to understand what the terms in these equations for velocity and acceleration in a rotating reference frame mean.
Is the problem even using a rotating frame? It looks to me like they are using a non-rotating frame, and the angular velocities and accelerations given are relative to that non-rotating frame.
 
  • #3
To me it looks like the particle is constrained to move in a circular groove cut into a square plate of side ##l##. The square plate has angular acceleration about one of its corners. So in addition to the contact forces exerted by the groove, the particle is subjected to the inertial forces in the plate's rotating frame.

The particle's angular velocity ##\alpha## (poor choice of symbol if you ask me) and angular acceleration ##\dot{\alpha}## are given relative to the center of the groove which is at distance ##\frac{l}{\sqrt{2}}## from the axis of rotation of the plate.
 
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Related to Question about Formulae for Motion in a Rotating Reference Frame

1. How do you calculate the Coriolis force in a rotating reference frame?

The Coriolis force in a rotating reference frame can be calculated using the formula F = -2mω x v, where F is the Coriolis force, m is the mass of the object, ω is the angular velocity of the reference frame, and v is the velocity of the object.

2. What is the formula for centripetal acceleration in a rotating reference frame?

The formula for centripetal acceleration in a rotating reference frame is a = ω^2r, where a is the centripetal acceleration, ω is the angular velocity of the reference frame, and r is the radius of the circular motion.

3. How do you calculate the centrifugal force in a rotating reference frame?

The centrifugal force in a rotating reference frame can be calculated using the formula F = mω^2r, where F is the centrifugal force, m is the mass of the object, ω is the angular velocity of the reference frame, and r is the radius of the circular motion.

4. What is the difference between tangential and radial acceleration in a rotating reference frame?

Tangential acceleration is the component of acceleration that is parallel to the velocity vector, while radial acceleration is the component that is perpendicular to the velocity vector. In a rotating reference frame, tangential acceleration is caused by changes in the speed of the object, while radial acceleration is caused by changes in the direction of the object's velocity.

5. How does the Coriolis force affect the trajectory of an object in a rotating reference frame?

The Coriolis force causes an object in a rotating reference frame to experience a deflection in its trajectory. In the Northern Hemisphere, the deflection is to the right of the object's intended path, while in the Southern Hemisphere, it is to the left. This effect is due to the rotation of the Earth and is responsible for the Coriolis effect in weather patterns and ocean currents.

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