Acceleration for a non-inertial reference frame

[Like Tony Stark]

Homework Statement

The disk with a slot rotates with ##\omega =20 \frac{rad}{s}## about a vertical axis that passes through its centre. There's a slider of mass ##0.73 kg## that oscilates because of the action of a spring (not shown in the picture) and it has a positive velocity ##90 \frac{cm}{s}## relative to the disk when it is in ##x=0##. Determine the normal force exerted by the slot on the slider when ##x=0##.

Relevant Equations

##a_I=a_o+\dot \omega \times r+\omega \times (\omega \times r)+2(\omega \times v_{rel}) +a_{rel}##.

Well, first a wrote the equation for acceleration in non inertial systems.

##a_I=a_o+\dot \omega \times r+\omega \times (\omega \times r)+2(\omega \times v_{rel}) +a_{rel}##.

Then, ##a_o=0## (because the system doesn't move), ##a_i=0## (because it is measured from the non inertial system), ##\dot \omega =0## (because the angular velocity is constant), ##r=0## (because ##x=0##)

So we get
##-2 (\omega \times v_{rel})=a_{rel}##

Then, we write Newton's equations
##x) Fe=m.a_{rel}##
##y) N-F_{Cor}=0##

Where ##Fe## is the elastic force and ##F_{Cor}=2(\vec \omega \times v_{rel})##

So I just have to replace with the numbers that I was given and get ##N##.

Are my ideas correct?

 

[kuruman]

There are two non-inertial frames, one fixed on the platform and one fixed on the slider. Which one are you considering? When you say that ##a_0=0## because the "system doesn't move" which system is this? What are you going to use for ##v_{rel}##? Also, don't replace any numbers before you get an expression for ##N## in terms of symbols all of which are given. That will make it easier to check your work.

 

[Like Tony Stark]

There are two non-inertial frames, one fixed on the platform and one fixed on the slider. Which one are you considering? When you say that ##a_0=0## because the "system doesn't move" which system is this? What are you going to use for ##v_{rel}##? Also, don't replace any numbers before you get an expression for ##N## in terms of symbols all of which are given. That will make it easier to check your work.

I'm considering a system fixed to the platform. I said that ##a_0=0## because the origin doesn't move. Then I said that ##a_i## is the acceleration measured from an inertial frame, but as I took the disk frame, I said that it was 0.

As for not replacing the numbers, yes, I'll take your advice, but I replaced them to check if the data that I plugged in was ok.

 

[kuruman]

So what's your symbolic answer?

 

[Like Tony Stark]

So what's your symbolic answer?

##y)N-\Sigma F_{rely}=m.a_{rely}##
##N=\Sigma F_{rely}+m.a_{rely}##

Where ##\Sigma F_{rely}## is the sum of the pseudo-forces acting on the ##y## direction and ##a_{rely}## is the acceleration of the slider relative to the disk.

 

[kuruman]

##y)N-\Sigma F_{rely}=m.a_{rely}##
##N=\Sigma F_{rely}+m.a_{rely}##

Where ##\Sigma F_{rely}## is the sum of the pseudo-forces acting on the ##y## direction and ##a_{rely}## is the acceleration of the slider relative to the disk.

Not what I had in mind. I am asking for an expression giving ##N## in terms of symbols for which you have numbers. You don't have numbers for any of the quantities on the right hand side except for the mass ##m##.

 

[Like Tony Stark]

Not what I had in mind. I am asking for an expression giving ##N## in terms of symbols for which you have numbers. You don't have numbers for any of the quantities on the right hand side except for the mass ##m##.

##y)N=m.a_{rely} -2(\omega \times v_{rel})##

##m##, ##\omega## and ##v_{rel}## are a given values; ##a_{rely}=0## because the slider just moves in ##x##

 

[kuruman]

Almost correct. You need to multiply the second term by the mass. Also, the problem gives only magnitude of ##\omega##, but I think it is safe to assume that it is in the ##+z## direction.