| Problem statement Solution video |
DISCUSSION THREAD
REMINDER: The system moves in a vertical plane.
Ask and answer questions here. You learn both ways.
DISCUSSION and HINTS
You need to break the problem into two parts: during impact and between post-impact and the maximum rotation angle of the bar. For the during impact portion, consider the following:
Step 1: FBDs
Draw a single free body diagram including the two pendulum masses, the bullet and the bar.
Step 2: Kinetics (linear impulse/momentum)
Is angular momentum for the system about point O conserved? If so, why? Energy is NOT conserved. Why?
Step 3: Kinematics
After impact, the velocity of the bullet is the same as the velocity of the lower pendulum mass. You need to relate the speeds of the two pendulum masses and the bullet to the angular speed of the bar using rigid body kinematics.
Step 4: Solve
From your equations solve for the velocities of the two pendulum masses and the bullet.
After impact, consider using conservation of energy for the system of the two pendulum masses and the bullet together.
How would you approach to solve for the maximum rotation angle? Is it something to do with breaking up the system to it being before and after impact and setting up the kinematic or something like that?
During impact, angular momentum is conserved, and mechanical energy is NOT conserved.
Following impact, angular momentum is NOT conserved, and mechanical energy is conserved.
Use those two observations to solve this problem in two parts: one DURING impact, and one AFTER impact.
Do we need to provide the answer of angular velocity as a vector?
Yes.
For the interval DURING impact, our first time should be immediately before impact right? And at that moment, we should assume the center of mass of the bullet is directly below O, right?
Yes.
Am I correct to assume that we should only use the x-component of the initial velocity of the bullet because that is the only things contributing to the angular momentum of the system? This is portion of the velocity is tangential to the path of the pendulum and thus impacts the pendulum's velocity, whereas the y-component is perpendicular to the path and thus has no effect on the pendulum's velocity.
If you try to find the angular momentum of the bullet about O immediately before impact, you'll discover that the y-component of the bullet's velocity will end up being irrelevant because it is in a cross product with r_b/O which only has a y-component as the bullet is directly below O, I believe. So at least for this case, the y-component of the velocity is irrelevant, but I wouldn't do this all the time.
Yes, that's correct because when you do the angular momentum equation solving for H1 = m(r x v), when written as vectors, the r cross v term will cancel out the j terms, pretty much leaving the equation as the x component of velocity times the distance from O to M.