- #1

Master1022

- 611

- 117

- 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

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

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