A mobile disk of radius R and mass M is moving above another immobile vertical disk of the same radius.
Initially the mobile disk is at rest at the highest point above the immobile disk and then it starts rolling without slipping. Assuming the mobile disk never slips, find the Θ for witch both disks are no longer in contact.
The Attempt at a Solution
This is what I tried:
First the CoM of Disk 1(mobile disk)is at distance 2R from the origin(for me the origin is the CoM of Disk 2(immobile disk)
The Disk 1 is going to be at distance 2RcosΘ when they are no longer in contact.
In the other hand, we know both objects will no longer be in contact when N=0
After working with Newton's Secod Law :
j)N-mgcosΘ=macp⇒ mgcosΘ=mv2/2R ⇒ (Vcm)2=2RgcosΘ
With this information I can apply conservation of energy:
Initially we have : mg2R
finally we are going to have: 1/2(Iw^2) + 1/2(m(vcm)2)) + mg2RcosΘ
⇒ mg2R = 1/2(MR^2)(Vcm2)/R2) + 1/2(m2RgcosΘ) + mg2RcosΘ
⇒ mg2R = 1/2(MR^2)2RgcosΘ/R2) + 1/2(m2RgcosΘ) + mg2RcosΘ
⇒ 2= cosΘ + cosΘ + 2cosΘ
⇒ Θ=1/2... but the right answer is Θ= 4/7 what am I doing wrong?