What Are the Conservation Laws in a Pendulum Collision Problem?

[Hanuda]

Homework Statement

A stick of mass m and length l is pivoted at one end. It is held horizontal
and then released. It swings down, and when it is vertical the free end elastically collides.
Assume that the ball is initially held at rest and then released a split second before the
stick strikes it

a. Which conservation laws apply here?
b. If the stick loses half of its angular velocity during the collision, what is the mass of
the ball, M?
c. Determine the speed V of the ball right after the collision

Homework Equations

Moment of inertia of the stick:

$$\frac{1}{3}mL^2$$

$$L=I\omega$$

$$\omega=\frac{v}{r}$$

$$K_{rotational}=\frac{1}{2}I\omega^2$$

The Attempt at a Solution

For (a), I said that conservation of energy, and conservation of linear/angular momentum also apply

For (b) I used conservation of energy just before the collision:

$$mg\frac{L}{2}=\frac{1}{2}I\omega^2$$

$$mg\frac{L}{2}=\frac{1}{2}(\frac{1}{3}mL^2)\omega^2$$

This gives me an omega of $$\omega=\sqrt{\frac{3g}{L}}$$

I then used conservation of energy before and after collision:

$$\frac{1}{2}I{\omega^2}=\frac{1}{2}I\frac{\omega^2}{4}+\frac{1}{2}M{V_{b}}^2$$

I reduce this down to: $$M{V_{b}}^2=\frac{3}{4}mLg$$

Then I used conservation of linear momentum (which is the part I'm not entirely sure about). I assumed the momentum of the mass of the stick before the collision was equal to the sum of the momentum of the CM and ball after the collision:

$$mv_0=mv_1 + MV_b$$
$$V_{cm}=\frac{L}{2}\omega=\frac{L}{2}\sqrt{\frac{3g}{L}}=\sqrt{\frac{3Lg}{4}}$$

==> $$\frac{mL\omega}{4}=MV_b$$

Isolating v and putting it into the $$M{V_{b}}^2$$ expression I get $$M=\frac{1}{4}m$$

Then the velocity of the ball comes out to be $$\sqrt{3gL}$$

Is this correct? For example, I did not use angular momentum, when my gut is telling me that I should have. But assuming that conservation of linear momentum holds, I don't need it right?

Many thanks for your help guys.

 

[voko]

For (b) I used conservation of energy just before the collision:

$$mg\frac{L}{2}=\frac{1}{2}I\omega^2$$

This is not conservation of energy. I recognize the right hand side as kinetic energy of the rod. The left hand side is probably the potential energy of the rod. Their sum would be the rod's total energy. But the equality makes no sense at all.

 

[Hanuda]

Hi Voko, why is that? I figured the potential energy of the rod when horizontal, is equal to the rotational kinetic energy of the rod when vertical, just before it strikes the ball. How is this not conservation of energy?

 

[voko]

I was misled by the "just before the collision" bit. Is the left hand side the initial potential energy?

 

[Hanuda]

The LHS is the initial potential energy, yes. The RHS is the rotational kinetic energy just before the stick collides with the ball.

 

[voko]

Then that part is correct.

 

[Hanuda]

But do I have the correct solutions for the velocity of the ball, and the mass M?

 

[voko]

I have not checked your algebra, but the general method appears correct.

 

[Hanuda]

As long as its correct conceptually, I'm on the right track. Thanks for your help!

 

[voko]

Regarding your doubt about linear vs angular momentum, you could redo the problem using the latter. You should get the same result.

 

[Hanuda]

Just for the sake of argument, how would you go about doing it for angular momentum? For example, I get the following expression for angular momentum:

$$I\omega=I\frac{\omega}{2} + MV_bL$$

But when I work this out, I get:

$$\frac{1}{6}mL\omega=MV_b$$

But when I used linear momentum, I got:

$$\frac{1}{4}mL\omega=MV_b$$

What went wrong?

 

[voko]

I have just realized that conservation of linear momentum does not hold here. This is because of the pivot, which prevents any linear motion of the rod.