- #1

Alettix

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## Homework Statement

A toy windmill consists of four thin uniform rods of mass m and length l arranged at rigth angles, in a vertical plane, around a thin fixed horizontal axel about which they can turn freely. The moment of intertia about the axel is ##I = \frac{4ml^2}{3}##.

Initially the windmill is stationary. A small ball of mass m is dropped from a height h above the end of a horizontal rod, with which it makes an elastic collision. What is the angular velocity of the windmill after the collision and with what velocity does the ball rebound?

## Homework Equations

Potential energy: ##E_p = mg\Delta h##

Kinetic energy: ##E_k = \frac{mv^2}{2} + \frac{I\omega^2}{2}##

Momentum: ##p=mv##

Angular momentum: ##L=I\omega##

## The Attempt at a Solution

The speed the speed the ball hits the windmill arm with is:

##v = \sqrt{2gh}##

Let ##u## be the velocity it rebounds with, and ##\omega## the angular velocity of the windmill.

Because the windmill cannot move translationally, translational momentum cannot be conserved. This means that an external force has to act at the axel to keep it back from accelerating downwards. Because and external force acts on the system, angular momentum is not conserved either (or is it? I am unsure about this, maybe it is conserved about the centre of the windmill, but I need some guidance here).

When the ball collides with the windmill it acts with an impulse, giving it an angular momentum:

## L_w = l \Delta p = l m (u-v)## (where ##u## is a negative quantity and hence the windmill will rotate downwards). This leads to:

## \omega = \frac{3(u-v)}{4l}##

If we now assume that the energy is conserved (elastic = totally elastic?):

## \frac{mv^2}{2} = \frac{mu^2}{2}+\frac{I\omega^2}{2}##

Inserting our previous expression for ##\omega## this equation gives:

## u = \frac{9-\sqrt{109}}{14} v## and hence: ##\omega \approx \frac{0.827v}{l}##.

The problem is that this is all wrong. The answer is supposed to be ##u = v/7## and ##\omega = \frac{6v}{7l}##. I am unsure about where my solution goes wrong. Is the assumption of energyconservation faulty? Is MoI conserved about the axel? But how can MoI be conserved about some points but not others?

Thank you for any help! :)