Collision with a pinned and unpinned rod on a frictionless surface

[n00bot]

1. The Problem: (Collision with a thin rod -- pinned and unpinned)
A small object with mass n and an initial velocity of v sticks to a long thin rod of mass m and length l. The motion takes place on a horizontal frictionless surface. Answer the questions for the situation where
a) the rod is pinned at its center (but able to spin frictionlessly), the object rebounds, and the coefficient of restitution is r.
b) The rod is not pinned and the object sticks to the rod.

2. The Questions:
1. In each situation, is linear or angular momentum conserved? Explain.
2. Determine the energy lost in the collision.
3. Where should the small object collide with the rod to maximize the loss of energy in the collision? Explain.


3. The Attempt at a Solution :
1. a) The pin applies force on the rod, so linear momentum is not conserved within the rod-object system (the pin transfers some momentum to the surface). Angular momentum is conserved since the rod is free to rotate.
b) Both angular and linear momentum are conserved: the rod is free to rotate, and free to move. Rotation and translation.

2. a) Elastic collision, so KE_initial = KE_final. I assume energy would be transferred to the rod from the object, but the system is constant. Maybe?

b) Since it is a completely inelastic collision, I know that KE_initial > KE_final. My best guess to determine loss of energy would be to set up a conservation of energy equation, and have m_obj-initialv_obj-initial² + 0 = (m_obj + m_rod)v_final². I'd then solve for v_final, and plug K_initial=m_obj-initialv_obj-initial²
K_final= (m_obj + m_rod)v_final Is that correct?


3. a) if KEinitial = KEfinal, then no energy would be lost, and it wouldn't make a difference where the rod was hit. But... what about the pin? (How could that be right?!)

b)I assume the max loss of energy would occur with a collision at the center, because that would cause the bar to rotate the least (or, not at all). I can't really figure out where to go with this, though, since there's no friction (so, where would the energy go?) How do I go about figuring this out?

Thanks

 

[annoymage]

hehe, I am new here, so i hope we all can work together.

well i think in

1. a)

linear and angular is not conserved.
reason: energy loss during collision (sound, heat, etc) and total initial energy is not equal to total final energy

1. b)

linear and angular in conserved, since is inelastic collision with no friction on surface

2. a) use the coeficient of restitution formula $$\textrm{C.O.R.} = \frac{|\vec{v_2} - \vec{v_1}|} {|\vec{u_2} - \vec{u_1}|}$$

substitute v1 and v2 in kinetic equation... the differences will get the energy loss.

2. b) there is no energy loss.

3. a) the edge of the rod, hmm, i think we can explain by relationship using formula of torque, force, and energy i guess

3. b) anywhere is the same, i think xP
anyway, please help me if i was wrong somewhere and I'm sorry if I'm wrong.. and i also sorry because i don't know whether i can leave my answer or not.. because i think only tutor can post answer here.. notify me if i did something wrong..
thanks

 

[kerimek]

for your question and for providing the necessary information to solve the problem.

1. In each situation, is linear or angular momentum conserved? Explain.
a) In situation a), linear momentum is not conserved within the rod-object system due to the presence of the pin. The pin exerts a force on the rod, causing a change in the linear momentum of the system. However, angular momentum is conserved since the rod is free to rotate without any external torque acting on it.
b) In situation b), both linear and angular momentum are conserved since there are no external forces or torques acting on the system. The rod is free to move and rotate without any external interference.

2. Determine the energy lost in the collision.
a) In situation a), the collision is elastic, so the initial and final kinetic energies are equal. Therefore, no energy is lost in the collision.
b) In situation b), since it is a completely inelastic collision, some energy is lost due to the deformation of the objects. The amount of energy lost can be determined by subtracting the final kinetic energy (which is zero) from the initial kinetic energy.

3. Where should the small object collide with the rod to maximize the loss of energy in the collision? Explain.
In order to maximize the loss of energy in the collision, the small object should collide with the rod at a point that will cause the rod to rotate the most. This would result in a larger deformation of the objects and thus a greater loss of energy. This point would be at the end of the rod, furthest from the pin, as this would cause the rod to rotate the most. However, since there is no friction in the system, the energy lost would not be significant. If there was friction present, the energy lost would be converted into heat due to the sliding motion between the objects.