Calculating speed and angular momentum

[bobie]

From conservation of kinetic energy we get:
$$\frac{i \omega _0^2}{2}+\frac{m v_0^2}{2}=\frac{i \omega _1^2}{2}+\frac{m
v_1^2}{2}$$
From conservation of angular momentum we get:
$$ i \omega _0+m r v_0=i \omega _1+m r v_1 $$
From conservation of linear momentum we get:
$$ mv_0 + mv_1 = Mv_r → 22 + 11.85 = 33.85 → v_r = 3.385 → E_k = 57.29 J $$

1) We know that the rod will not react in

.2)The equations are the rule and they are the way to predict the behavior.
I am not sure what you mean by "resistance".
.

By resistance I mean that the rod reacts like a ball of mass of 10/3
1) the rod has reacted in the same way 10/3 both with and without fulcrum.
2) I know I am asking too much, but as they found that a rod has I = 1/12, they might have studied that 1/12 corrsponds to 1/3 of the real mass in the collision. That would have made things easier and we might have got immediately $v_1 = 7/13 v_0 = 11.85 , v_r = 20/ (13[*M]) = 3.385$

 

[jbriggs444]

What about the contribution to kinetic energy due to the rod's final linear motion? What about the contribution to momentum due to the ball's final linear motion?

The naming convention for the variables needs thought as well. It was one thing to talk about $v_0$ and $v_1$ when we only had an initial and a final velocity of the ball. Now we have an initial and final velocity of the rod as well. I am a firm believer in documenting variable names.

 

[Dale]

From conservation of kinetic energy we get:
$$\frac{i \omega _0^2}{2}+\frac{m v_0^2}{2}=\frac{i \omega _1^2}{2}+\frac{m
v_1^2}{2}$$

You are forgetting the term for the translational KE of the rod. If we denote the velocity of the center of mass of the rod as $V$ (to go along with the mass of the rod as $M$) then that will be $\frac{1}{2} M V^2$. With $V_0$ before the collision and $V_1$ after the collision.

From conservation of linear momentum we get:
$$ mv_0 + mv_1 = Mv_r $$

This is not correct. The conservation of momentum relates the momentum before the collision (subscript 0) to the momentum after the collision (subscript 1). On the left hand side you should have two terms, one expressing the linear momentum of the particle (lower case m and v) before the collision and the other expressing the linear momentum of the rod (capital M and V) before the collision (subscript 0). On the right hand side you should have two terms, one expressing the linear momentum of the particle (lower case m and v) after the collision and the other expressing the linear momentum of the rod (capital M and V) after the collision (subscript 1). All of the subscript 0's should be on the left, and all of the subscript 1's should be on the right. There should be no subscript r anywhere.

By resistance I mean that the rod reacts like a ball of mass of 10/3

This concept of resistance is not a standard concept. Let's drop it. It is just a distraction and is not necessary for determining the physics.