- #1
person123
- 328
- 52
Homework Statement
There are two discs with equivalent density and thickness. One has radius r1 while the other has radius r2. r2 is twice as great as r1 The larger disc has an initial angular velocity ω. The two discs then come in contact with one another and friction causes them to rotate with the same linear velocity. What are the angular velocities of the two discs?
For everything related to the larger disk, besides the initial angular velocity which is just ω, I'll subset it with 2. For everything related to the smaller disk, I'll subset it with 1.
Homework Equations
##v=ωr##
##KE=1/2Iω^2##
##I_{disc}=1/2mr^2##
##E_{i}=E_{f}##
The Attempt at a Solution
##½I_{1}ω^2=½I_{1}ω_{1}^2+½I_{2}ω_{2}^2##
##½(½m_{1}r_{1}^2)ω^2=½(½m_{1}r_{1}^2)ω_{1}^2+½(½m_{2}r_{2}^2)ω_{2}^2##
##r_{1}^4ω^2=r_{1}^4ω_{1}^2+r_{2}^4ω_{2}^2##
Since ##r_{2}=2r_{1}##:
##r_{1}^4ω^2=r_{1}^4ω_{1}^2+16r_{1}^4ω_{2}^2##
By knowing that ##v=ω_{1}r_{1}=ω_{2}r_{2}⇒ω_{1}=2ω_{2}## I could solve for the angular velocity of the two discs.
For ##ω_{1}##:
##r_{1}^4ω^2=r_{1}^4ω_{1}^2+16r_{1}^4\frac{ω_{1}^2} 4##
##r_{1}^4ω^2=r_{1}^4ω_{1}^2+4r_{1}^4ω_{1}^2=5r_{1}^4ω_{1}^2##
##ω^2=5ω_{1}^2⇒ω_{1}=\frac {ω} {\sqrt5}##
For ##ω_{2}## :
##r_{1}^4ω^2=r_{1}^4(2ω_{2})^2+16r_{1}^4ω_{2}^2##
##r_{1}^4ω^2=4r_{1}^4ω_{2}^2+16r_{1}^4ω_{2}^2=20r_{1}^4ω_{2}^2##
##ω^2=20ω_{2}^2⇒ω_{2}=\frac {ω} {2 \sqrt{5}}##
When I posted these answers on Walter Lewin's youtube channel, he posted it which means it's incorrect. I've checked it several times but I can't find any errors. Does anyone know where I went wrong?
Last edited: