- #1

VSayantan

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## Homework Statement

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A mass ##m## travels in a straight line with velocity ##v_0## perpendicular to a uniform stick of mass ##m## and length ##l##, which is initially at rest.. The distance from the center of mass of the stick to the path of the traveling mass is ##h##. Now the traveling mass ##m## collides elastically with the stick, and the centre of the stick and mass ##m## are observed to move with equal speed ##v## after the collision. Assuming the traveling mass can be trated as a point mass, and the moment of inertia of the stick about its centre is ##I=\frac {ml^2}{12}##, calculate the distance ##h##.

## Homework Equations

For

*elastic collisions,*

I. Kinetic Energy is conserved.

II. Linear Momentum is conserved.

Additionally, since there is no external torque about the centre of mass

III. Angular Momentum about the centre of mass is conserved.

$$((E_k)_{ball})_i={\frac 1 2}m{v_0}^2$$

$$((E_k)_{ball})_f={\frac 1 2}mv^2$$

$$((E_k)_{stick})_i=0$$

$$((E_k)_{stick})_f={\frac 1 2}mv^2 + {\frac 1 2}I\omega^2$$

$$((\vec p_{ball})_i=mv_0 \hat i$$

$$((\vec p_{ball})_f=mv \hat i$$

$$((\vec p_{stick})_i=0$$

$$((\vec p_{stick})_f=mv(\hat i cos \theta + \hat j sin \theta)$$

$$((\vec L_{ball})_i=\vec{r} \times m\vec{v_0}$$

$$((\vec L_{ball})_f=0$$

$$((\vec L_{stick})_i=0$$

$$((\vec L_{stick})_f=I\vec \omega + ?$$

## The Attempt at a Solution

Conservation of Linear Momentum gives

$$\sum {\vec p}_i=\sum {\vec p}_f$$

$$\Rightarrow mv_0 \hat i=mv\hat i + mv(\hat i cos \theta + \hat j sin \theta)$$

Equating coefficients of ##\hat i## and ##\hat j## -

$$v sin \theta =0$$

$$v_0=v(1+cos \theta)$$

Taking square and adding and simplifying

$$2v=v_0$$

From Conservation of Kinetic Energy

$$E_i = E_f$$

$$\Rightarrow {\frac 1 2}m{v_0}^2={\frac 1 2}mv^2+{\frac 1 2}mv^2 + {\frac 1 2}I\omega^2$$

After simplification, this gives

$$m({v_0}^2 - v^2)=I{\omega}^2$$

Using $$2v=v_0$$, this gives

$$I{\omega}^2=2mv^2$$

But I cannot calculate the angular momentum of the stick.

Is it $$((\vec L_{stick})_f=I\vec \omega + m\vec v ({\frac l 2}-r)$$

If someone has any suggestion, it might be helpful.

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