FIGURE P12.82 shows a cube of mass m sliding without friction at speed v₀. It undergoes a perfectly elastic collision with the bottom tip of a rod of length d and mass M = 2m. The rod is pivoted about a frictionless axle through its center, and initially it hangs straight down and is at rest. What is the cube's velocity—both speed and direction—after the collision?

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Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem or a science fair project. A student uses a billiard ball of 0.16 kg and a cue stick of 0.48 kg to show rotational collisions. The ball rolls on the table and hits with a speed of 2.0 m per second. The edge of a 1.5 m homogeneous QE stick hanging at rest straight down the cue stick rotates about a frictionless axle passing through the stick's center, calculate the speed of the ball after the impact assuming a perfectly elastic collision. OK. So that is our end goal. So we're given some multiple choice answers. They're all in the same units of meters per second. So let's read them off to see what our final answer might be. A is zero. B is 1.0 C is 2.0 and D is 4.0. OK. So first off, let us consider that the billiard ball and the acoustic are part of a system and we will focus on that, we will focus on to solve this problem. So the bill, your ball and the cue stick is our main focus to help us solve this problem. And it's in a system. So let us also assume that a perfectly elastic collision occurs because the problem states. So thus the energy and the angular momentum where no external torque is applied to. So where there's no external torque applied, reiterate that will be conserved. So the angular momentum will be conserved. So the law of the conservation of energy states that the initial energy is equal to the final energy. So we can state that let's call it equation one, we can state that one half MB the mass of the billiard ball or the mass of the ball multiplied by the initial velocity squared is equal to one half the mass of the ball multiplied by the final velocity squared plus one half the moment of inertia for the cue stick multiplied by the angular speed of the acoustic after the collision square. OK. And V I is the initial velocity of the ball and VF is the final velocity of the ball. OK. So note that the equation to determine the moment of inertia for the Q stick about an axis axis passing through the center of mass is written as and then we need to treat the Q stick as a rod. So let's call this equation two. So the moment of inertia for the Q stick, which I may have denoted as I subscript S is equal to one divided by 12, multiplied by the mass of the Q stick. I'm gonna denote it as M subscript S multiplied by the moment of inertia of the queue stick squared. So note that the mass of the stick is three times the mass of the ball. So we can write that MS is equal to three MB. And let's call this equation three. OK. So now we need to combine the equations 12 and three. So let's scroll down a little bit to give us some space. OK. So one half the mass of the ball multiplied by the initial velocity squared is equal to one half multiplied by the mass of the ball multiplied by the final velocity squared plus one half multiplied by one divided by 12, 1 12, multiplied by three MB, multiplied by the inertia or MB is the mass of the ball and then is, is the moment of inertia for the acoustic squared multiplied by the angular speed of the acoustic after the collision squared. And let's call this equation four fantastic. So now we must use the conservation of angular momentum to find the relationship between omega which is the angular speed, the initial velocity V I and the final velocity VF OK. So the initial angular momentum is written as L subscri il I, this is the initial angular momentum is equal to the mass of the ball multiplied by the initial velocity multiplied by the moment of inertia of the acoustic divided by two. And this is the equation five. OK. So the final angular momentum is written as LF so the final angular momentum is equal to the mass of the ball multiplied by the final velocity multiplied by the moment of inertia. For the cue stick divided by two plus the moment of inertia multiplied by the angular speed. Or we could write it in this quality equation six at the final angular momentum is equal to the mass of the ball multiplied by the final velocity multiplied by the moment of inertia of the acoustic, divided by two plus 12, multiplied by the mass of the Q stick multiplied by the moment of inertia acoustic squared multiplied by the angular speed of the acoustic after collision. OK. So now we must apply the conservation of momentum which states that L I, the initial angular momentum equals the final angular momentum. So when we apply that we should get that the mass of the ball multiplied by the initial velocity multiplied by the inertia of the acoustic. Divided by two is equal to the mass of the ball multiplied by the final velocity multiplied by the inertia of the Q stick divided by two plus 1 12 multiplied by the mass of the Q stick multiplied by the inertia of the acoustic squared. Multiplied by the angular speed. So let's simplify that. So we get the mass of the ball, the mass of the ball multiplied by the initial velocity multiplied by the inertia of the acoustic is equal to the mass of the ball multiplied by the final velocity multiplied by the inertia. The acoustic divided by two plus 1 12. Or I should say when we simplify these, we need to plug in three MB for the mass of the Q stick. So 1 12 multiplied by three multiplied by MB multiplied by the moment of inertia or the Q stick squared multiplied by the angular momentum. OK. So now we could write that V one equals V two plus one half f sorry one half multiplied by the moment of inertia for the QE stick multiplied by the angular speed. So this is the equation seven. So now we need to rearrange this equation to solve for the angular speed, which is represented by omega. So Omega when we isolate and solve for Omega only using a little bit of algebra, we get two multiplied by the initial velocity minus the final velocity divided by the moment of inertia for the acoustic. So now we need to substitute equation seven into equation four. And when we do that, we get that V I, the initial velocity squared is equal to the final velocity squared plus one half multiplied by the moment of inertia of the Q six squared multiplied by two multiplied by the initial velocity minus the final velocity divided by the moment of a nurse of the Q stick all squared. So when we simplify that we get that the initial velocity is equal to the final velocity. So the initial velocity squared is equal to the final velocity squared plus the initial velocity squared. What the final velocity squared minus the initial velocity multiplied by the final velocity. OK. Now is a lot, but we're almost done. So now we need to set this equal to zero. So two multiplied by the final velocity squared minus two, multiplied by the initial velocity multiplied by the final velocity is equal to zero. So we can say that the final velocity multiplied by, we simplify more multiplied by the final velocity minus two, multiplied by the initial velocity is equal to zero. So there are two solutions. Now, the first solution is the final velocity is equal to two multiplied by V I. But this answer isn't correct because the stick was motionless before impact. And our second equation is VF or something you could call it V 22, but it's equal to 0 m per second. So as we know this answer was, write it in blue is can't is like can't be possible because the stick was motionless before impact. So after impact, so VF equals 0 m per second. So after impact, the ball will move or go to rest since the kinetic energy is transferred to the Q stick. So this is the correct answer. 0 m per second. Hooray, we did it. So let's go look at our multiple choice answers to see what the correct answer is. So the correct answer has to be a 0 m per second. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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Key Concepts

Conservation of Momentum

Elastic Collision

Rotational Dynamics