Angular momentum theory problem

[charbon]

Homework Statement

Verify the isomorphism between rotational and translational movement (This is for writing the theory section of my lab report. I should also add that this is using rotating disks)

Homework Equations

\vec{p}=m\vec{v}
\vec{L}=I\vec{\omega}
I=1/2MR^2
M = the mass of the disk
R = the radius of the disk

The Attempt at a Solution

L = \vec{R}X\vec{p}
p = m\vec{v}
\vec{v}=d\vec{R}/dt
\vec{v} is perpendicular to \vec{R} therefor
L=mR^2ω
since p = mRω

I'm off by 1/2 of the other definition of L ( =Iω).
Am I doing something wrong?

Thanks in advance for the replies!

Edit: sorry for the english, it's not my primary language!

 

[member 553083]

The attempt at a solution is incomplete and incorrect. To verify the isomorphism between rotational and translational motion, we need to consider both Newton's second law of motion and the conservation of angular momentum equations. Newton's second law of motion states that the acceleration of an object is proportional to the net force acting on it: F = ma where F is the net force, m is the mass of the object, and a is the acceleration. The conservation of angular momentum equation states that the angular momentum of an object is constant as long as there are no external torques acting on it: L = Iω where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. By comparing the two equations, we can see that the moment of inertia (I) is directly proportional to the mass (m). This is the key to understanding the isomorphism between rotational and translational motion. To further illustrate this isomorphism, we can consider a rotating disk with radius R and mass M. The moment of inertia of this disk is given by the equation: I = 1/2MR2 We can also consider the translational motion of this disk. By substituting the mass and radius of the disk into Newton's second law of motion, we can see that the acceleration of the disk is directly proportional to the angular velocity of the disk (ω): a = mRω Therefore, we can conclude that the rotational and translational motion of a rotating disk is isomorphic, meaning that they are equivalent in terms of their effect on the object's acceleration and angular momentum.