Solve Rotational Physics Problems: Circular Disk & Squirrel Example

[someone1]

Hey, i have this big packet filled with questions i don't know the answers to, but me and my friends have been able to knock out all but three, so i figured i'd ask the brilliant minds that make up this great community for help ;)



A hungry squirrel of negligible size (mass m= 2.0 kg) is sitting on the edge of a circular disk of radius R = 2.0 m and mass M=3.0kg. The disk is rotating frictionlessly with a rotational velocity wo=1.0 rad/sec. At the center of the disk is placed a walnut and the squirrel tries to reach the center to get it.

A) Suppose the squirrel moves radially toward the center. When he comes to the distance r=1.0m from the center, he stops. What is the angular velocity of the disk?

B) At r=1.0m, what is the centripetal acceleration of the squirrel?

C) Suppose the static coefficient of fiction between the squirrel and the disk is uS=.5. At r=1.0m, does the squirrel slip?



There's one of the three for you guys to munch on, I'll post the others if help on this can be given (they are all about Rotation). (not answers per say, just how to approach the problem, formulas to use, we REALLY REALLY don't know how to do this so any and all help would be appreciative!).



Thanks in Advanced!

 

[Meir Achuz]

For (a), use cons of anglar momentum
L=Iw+(mr^2)w.

 

[someone1]

Thanks for the speedy reply, but can you give an explanation for what each variable is? I think we may be misreading the formula :/

L is angular momentum, correct?
r is radius (in this case 1m)?
m would be Mass? but of what, the disk or squirrel?
unsure as to what Iw and w are.

also note, it is a CIRCULAR DISK if you haven't already!

 

[daniel_i_l]

L -> Total angular momentum
I -> Moment of inertia of disk
w -> angular velocity of disk
m -> mass of squirrel
r -> distance of squirrel from center
The first term is the AM of the disc and the second one is the AM of the squirrel, the sum of these two (L) is conserved.

 

[someone1]

AM of the disc? what does that mean?

(I think maybe i need the lamnest terms for all of this... or even be babied through it all... but i have done rotational problems of more basic caliber)

 

[hage567]

AM means Angular Momentum. Is that what you don't get?

 

[someone1]

I've been trying to use that formula for hours now and I'm not sure if its the right formula to use. It doesn't even seem to apply to this problem :/

Any other ideas?

 

[someone1]

we kinda used our logic to figure these out, but we're not sure about the answers... here's what we have:
a) w= v/r
v=rw
v = 2(1)
v = 2m/s
w=v/r
w=2/1
w=2m/s
w=2rad/sec

b)ac = v^2/r = 4/1
4m/s^2

i'm not sure if that's right or not, can some verify that for me?

 

[hage567]

No, that doesn't look right. I'm not sure what you are trying to do here. As Meir said above, you need to use conservation of angular momentum. So you need to find the initial angular momentum when the squirrel is at the edge of the disk. After the squirrel moves into 1 m, you must find a new expression for the final angular momentum. Note that to conserve angular momentum the angular velocity must increase when the squirrel moves inward. You must consider the moment of inertia of the squirrel + the disk. So do you know what the moment of inertia is for a point mass (squirrel) and cylindrical disk? (See post #2 actually)
Don't change things into tangential velocity for part a), it's not necessary.

 

[someone1]

Does this make more sense?
For part a:
I1W1 = I2W2 ... which equals... m(r1^2)w1= m(r2^2)w2... and then the m's cancel.. and you solve for w2... so... w2= w1(r1^2/r2^2)... which is 1(2^2/1^2) = 4

 

[hage567]

You're getting the right idea, but you're STILL not calculating I correctly! You must consider the moment of inertia of the squirrel AND the disk together, so $$I=I_s+I_d$$. Take the squirrel as a point mass, with $$I_s=M_sR_s^2$$, and the disk as $$I_d=0.5M_dR_d^2$$. The mass of the squirrel is not the same as the mass of the disk. So for the initial angular momentum, you should get

$$L_i= I\omega_i$$
$$L_i=(M_sR_s^2 + 0.5M_dR_d^2)\omega_i$$

Here, R is 2 meters for both Rs and Rd, since the squirrel is at the edge of the disk.

Can you finish it now? For the final angular momentum, Rs will be 1 m since the squirrel has moved, while Rd will remain at 2 m. Then equate your equation for the final angular momentum to the initial angular momentum, and solve for w2. Note: the masses will NOT cancel out.