[0:27]Hi, grade 12s, I'm Joyce Phooko. And I'm John McBride. The topic today is the principle of conservation of linear momentum. The principle of conservation of linear momentum states that the total linear momentum of an isolated system remains constant. This means that the total linear momentum of an isolated system is conserved. One of the best ways to demonstrate the effects of this principle is to play with this apparatus which we call Newton's cradle. Joyce, please lift the one ball and release it.
[1:05]The first ball collides with the next ball, which in turn collides with the next and the next until the last one, which swings up and out. Let's do that again, and watch closely what happens.
[1:23]When one ball is lifted up and released, it strikes the first one of the other four. Each ball in turn passes the impulse to the next ball until we get the last ball which moves up and away. It receives the momentum and it has nothing to stop it from swinging up and away. And you will see that the motion repeats itself from the other side. We can calculate the total momentum before the collision. Let the mass of each of the balls be M. The velocity of the ball on the right is V, moving to the left. The velocity of all the other balls is zero. The total momentum before the collision is MV towards the left. After the collision, the ball on the right stops moving. All the other balls are also stationary. So, all these four balls have no momentum, but the ball on the left move with the velocity V to the left. The total momentum after the collision is MV towards the left. The total momentum before the collision equals the total momentum after the collision. The principle of conservation of momentum applies to Newton's cradle because it is an isolated system. No external forces act on the system during the collisions. What is even more amazing is this: When we let two balls collide with the other three, the two balls swing up on the other side. And it's the principle of conservation of linear momentum that holds true. Today, our task is to investigate whether the principle of conservation of linear momentum holds true for an explosion, which, as you know it, is inelastic. Two trolleys are placed on a track and released by a spring. They move in opposite directions.
[3:29]Before the explosion, both trolleys are at rest. They have zero momentum and zero kinetic energy. After the explosion, they have kinetic energy and momentum. Kinetic energy is not conserved during an explosion, therefore an explosion is inelastic. The question we are investigating is: Does the principle of conservation of linear momentum apply to isolated systems during inelastic collisions and/or explosions? The apparatus for this investigation is an almost frictionless track, two trolleys and some extra mass pieces. Two stoppers at each end of the track, measuring tape, a ruler and a spirit level. The trolleys have a spring-loaded mechanism which pushes them apart when you hit the trigger with something flat like a ruler. Like this.
[4:28]To ensure that the explosion takes place in an isolated system, we have to make sure that the track is absolutely level. If the track slopes one way or the other, the trolleys will be influenced by their weight pulling them to either the left or the right. The system will not be isolated. When the track is perfectly level, both trolleys remain stationary at any position on the track. This is an isolated system. So the most important procedure in the investigation is the leveling of the track. If you have a spirit level available, you can use it to help you level the track. Place the spirit level lengthways on the track around the center. Adjust the leveling screws under the track until the bubble on the spirit level is in the center. Now move the spirit level up and down the track to check that at every position the track is correctly aligned. Now, you need to check that the horizontal alignment of the track is correct. Place the spirit level across the track and adjust the leveling screws or supports until the bubble is again in the center. Move the spirit level to various positions on the track to check and adjust the horizontal adjustment. The real test comes now. Even if you don't have a spirit level, this is the way you finally know that your track is level. If the track is level, the trolley remains stationary. The trolleys remain stationary, no matter where it is positioned on the track.
[6:43]Okay, so now you know that we have to get these trolleys to explode apart from each other and to hit their stoppers at exactly the same time. Let's try it first with two trolleys of equal mass. It makes sense that the trolleys of equal mass accelerate at the same rate when the same force is exerted on them. That is what Newton's second law tells us. So, the two trolleys will have the same final velocity as they travel down the track in opposite directions. They hit the barrier at the same time if they travel the same distance. So we position the two trolleys of equal mass in the center of the track. Measure the distance from the front of each trolley to the stopper. And we release the trolleys from their positions. The mass of each trolley equals 250 g. The distance of the red trolley to the stopper is equals to 63 cm. The distance of the blue trolley to the stopper is equals to 63 cm. And we release the trolleys, listen carefully. The trolleys must hit their end stoppers at exactly the same moment.
[8:44]They hit the stoppers at exactly the same time. Now what would happen when one of the trolleys is twice the mass of the other trolley? Let's make the red trolley twice the mass of the blue trolley.
[9:03]We need to find a position which allows them to both hit their respective barriers at exactly the same time. We could try endless possibilities by trial and error, or we can work out the correct starting position for these trolleys. We can use the principle of conservation of linear momentum. Applying the principle of conservation of momentum, we know that the initial linear momentum of the system is equal to its final momentum. Therefore, 2mv red is equal to -mv blue.
[9:39]Dividing through by the mass M, we get 2V red equals -V blue. But delta T is the same for both trolleys, therefore, 2 delta X red is equal to delta X blue. Delta X red is equal to half delta X blue. So, the red trolley has twice the mass of the blue trolley. Therefore, the red trolley accelerates at half the rate of the blue trolley. It covers half the distance of that the blue trolley covers in the same time. We need to divide the track into thirds. The trolley of the greater mass will travel 1/3 of the track while the trolley of the lesser mass will cover two thirds of the track so that they both hit their stoppers at the same time. Right, let's position the trolleys more carefully and we measure that distances from their stoppers.
[10:45]The mass of the red trolley is 500 grams and that of the blue trolley is 250 grams. The distance of the red trolley from its stopper is 42 cm and the distance of the blue trolley from its stopper is 84 cm.
[11:10]Perfect. They hit the stoppers simultaneously at exactly the same time. Now we will try one trolley with three times the mass of the other. Where should we position the trolleys at the start? Applying the principle of conservation of momentum, we know that the initial linear momentum of the system is equal to its final momentum. Therefore, 3mv red equals -mv blue. Dividing through by the mass M, we get 3V red equals -V blue. Delta T is the same for both trolleys, therefore, 3 delta X red is equal to delta X blue. So, delta X red is equal to 1/3 of delta X blue. So, let's add the extra mass piece to the red trolley.
[12:11]The distance covered by the red trolley is 1/3 of the distance covered by the blue trolley in the same amount of time. This time, we divide the track into four equal parts. The red trolley must travel 1/4 of the track and the blue trolley travels 3/4 in the same length of time. Position the two trolleys at 1/4 of the length of the track with the red trolley having the least distance to travel. Let's position the trolleys as accurately as we can, and we measure their distances from their stoppers. The mass of the red trolley is 750 grams and that of the blue is 250 grams. The distance of the red trolley from its stopper is 31 cm and the distance of the blue trolley from its stopper is 93 cm. Let's release the trolleys and listen to them crashing into their stoppers.
[13:16]Perfect. They hit the stoppers simultaneously at exactly the same time. Right, we have sufficient evidence now to come to a conclusion about whether the principle of conservation of momentum applies to these explosions of the trolleys. You've recorded the results, so now it's your task to complete the report on this investigation and to hand it in to your teacher.
