Rotational Motion & Rotational Kinetic Energy

[robert6774]

Four children stand at the edge of a circular horizontal platform that is free to rotate about a vertical axis. Each child has a mass of 35 kg and they are at positions that are a quarter-circle from each other. The platform has a moment of inertia equal to 500 kg*m^2 and a radius of 2.0 m. The system is initially rotating at 6.0 rev/min. The children walk towards the center of the platform until they are 0.50 m from the center.

m= 35 kg
I= 500 kg*m^2
r1= 2.0 m
w(initial)= 6.0 rev/min or 0.628 rad/s (correct me if I'm wrong)
r2= 0.50 m

(a) What is the rotational speed of the platform when the children are at the 0.50 m positions?

(b) What is the change in kinetic energy of the system?

Here are a few equations I might need to use:

w= d(theta)/dt

a= dw/dt

w= w(initial)+at

(theta)= w(initial)t + 1/2at^2

2a(theta)= w^2 - w(initial)^2

s=r(theta)

a= w^2r

K= 1/2 Iw^2

I don't think I should bring time as a variable into the picture because it would make it more things more complicated. I don't really know where to start. The only thing I've really done so far is write down the givens and draw a diagram. Thats a lot of equations but I know I don't need them all.

I tried using a= w^2r but I ended up with weird units. Maybe I don't understand the equation. So it would be (6.0)^2(2.0)= 72 rev/min*m. What?

The thing is I feel as though I'm missing one too many variables. When I try to utilize an equation, I can't solve it. For example, when I try to use w= w(initial)+at I'm missing both the acceleration and the time. When using 2a(theta)= w^2 - w(initial)^2 I'm missing the acceleration and theta.

Because my only lead with a= w^2 isn't working, I'm stuck.

Any help and guidance would be very much appreciated, thank you.

 

[tiny-tim]

welcome to pf!

hi robert6774! welcome to pf!

(have a theta: θ and an omega: ω and an alpha: α and try using the X² and X₂ icons just above the Reply box )

Four children stand at the edge of a circular horizontal platform that is free to rotate about a vertical axis. Each child has a mass of 35 kg and they are at positions that are a quarter-circle from each other. The platform has a moment of inertia equal to 500 kg*m^2 and a radius of 2.0 m. The system is initially rotating at 6.0 rev/min. The children walk towards the center of the platform until they are 0.50 m from the center.…

I tried using a= w^2r …

you don't need the force or acceleration or time

find ω_f by using conservation of https://www.physicsforums.com/library.php?do=view_item&itemid=313" ), applied to the children-and-platform

 

[robert6774]

Oh wow now I feel stupid. One of those "oh...duh" moments. Anyway thank you very much for your help.

One thing though, is my conversion from 6.0 rev/min to 0.628 rad/sec correct? I tried using the latter in the conservation of angular momentum equation but I ended up getting a slower second velocity which doesn't make sense.

 

[tiny-tim]

hi robert6774!

(just got up :zzz: …)

6.0 rev/min = 6.0/60 rev/sec = 0.1 rev/sec = 0.1*2π rad/sec = 0.628 rad/sec seems ok

how did it come out slower?

 

[robert6774]

I'm still getting the wrong answer. The key says its 12 rev/min but I'm getting 8.21 rev/minute.

Here's my work:

Conservation of angular momentum (of children and platform) gives:

I$$\omega$$$$_{}1$$+m$$\omega$$$$_{}1$$r$$_{}1$$ = I$$\omega$$$$_{}2$$+m$$\omega$$$$_{}2$$r$$_{}2$$

Solve for $$\omega$$$$_{}2$$,

$$\omega$$$$_{}2$$= ((I+mr$$_{}1$$)$$\omega$$1)/(I+mr$$_{}2$$)

Plug in values:

((500+(4)(35)(2.0))(6.0))/(500+(4)(35)(0.5)) = 4680/570 = 8.21 rev/min

Where did I go wrong?

I'm pretty new to the Latex stuff so this attempt at making the equations look pretty may not go so well, bear with me please.

 

[tiny-tim]

hi robert6774!

(please don't mix latex and text in the same line, it's very difficult to read )

you're using mωr for angular momentum, it has to be radius "cross" momentum, = r x mv = mωr²