- #1

Nexus99

- 103

- 9

- Homework Statement
- A rigid rod OA of negligible thickness, mass M and length L, with linear density ## \lambda (r) = kr ## (where r is the distance from the extreme O), lies in a smooth plane and is initially stationary, hinged in its extreme O. At the time ##t = t_0## a material point of mass m = 2M, which moves on the plane with speed ## v_0 ## perpendicularly directed to the rod, hits it in the extreme A in a completely anaelastic way. Calculate: 1) the moment of inertia of the rod as a function of M and L, evaluated with respect to an axis perpendicular to the plane and passing through O. 2) The angular velocity of the system after the collision 3) The moment of inertia of the system after the impact with respect to the previous axis 4) The distance of the new center of mass after the impact from the point O 5) The magnitude of the vincular reaction which acts after the collision

- Relevant Equations
- Center of mass, moment of inertia, kinetic energy

I'm struggling doing point 5, i have no idea how to solve that question. In point 1 i obtained the following result:

## I=\frac{ML^2}{2}## calculating the integral of dI, the infinitesimal moment of inertia of a small section of the rod of length dr.

2) Through the conservation of angular momentum (calculating angular momentum in point O) i obtained

## ω= \frac{4v0}{5L} ##

3) i obtained through huygens steiner:

## I=\frac{5ML^2}{2} ##

4) The center of mass before the collision is: ## x_{CM}=\frac{2L}{3} ## . After the collision:

## x_{CM}=\frac{8L}{9} ##

5) i don't know

## I=\frac{ML^2}{2}## calculating the integral of dI, the infinitesimal moment of inertia of a small section of the rod of length dr.

2) Through the conservation of angular momentum (calculating angular momentum in point O) i obtained

## ω= \frac{4v0}{5L} ##

3) i obtained through huygens steiner:

## I=\frac{5ML^2}{2} ##

4) The center of mass before the collision is: ## x_{CM}=\frac{2L}{3} ## . After the collision:

## x_{CM}=\frac{8L}{9} ##

5) i don't know

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