Episode 19 explores the concept of center of mass, and specifically how the momentum of the center of mass is affected during collisions of the objects in the system.
Episode 19 explores the concept of center of mass, and specifically how the momentum of the center of mass is affected during collisions of the objects in the system. We start by observing a child and parent playing on a seesaw at the park (0:41). The center of mass calculation is presented, and an example provided (1:22). The action is quickly relocated to an ice rink and the collision of two ice hockey players to see that the momentum of the center of mass is conserved (4:23).
The Question of the Day asks: (6:17)
Two friends attempt to balance on a paddleboard of length L. One friend is 3 times as massive. If the heavier friend stands on one end of the board, where should the lighter friend stand to maintain balance?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on finding the center of mass of a system.
Let’s Zoom out:
Unit 5 – Momentum
Topic 5.1-5.4
Big Idea – Force Interactions, Change, and Conservation
A child and her dad head to the playground to try out the seesaw. The child takes her place on one end of the seesaw, and her dad takes his seat on the opposite end. It was the least fun trip to the park that the child ever had.
Let’s Zoom in:
As it turns out, the center of mass of the child - dad system was not over the fulcrum of the seesaw, and as a result the child sat high in the air with her feet dangling above the earth anytime her dad would sit down. The center of mass is essentially a weighted average that is dependent upon both mass and distance. For the sake of simplicity, we will only be dealing with one-dimensional systems when finding the center of mass.
To calculate the center of mass, it is important that we pick an origin. I like to pick one end to be the origin and make all measurements of distance from that location. Once the origin is selected, we take the sum of each object’s mass multiplied by its distance from the origin. We then divide that sum by the mass of the system.
For our dad and his daughter attempting to use the seesaw, let's say that he has a mass of 75 kg while his daughter has a mass of 15 kg, and the length of the seesaw board is 6m. If she is sitting at the origin, and the dad is sitting on the opposite end 6 m away, then the center of mass for the child-dad system would lie much closer to the dad than his daughter. The dad’s mass (75 kg) multiplied by his distance from his child (6 m) divided by the total mass of 90 kg would yield a center of mass for the system of 5 m from the origin, or 1 m from the dad’s location.
The Physics 1 exam won’t expect you to necessarily calculate the center of mass, but you will certainly be expected to understand the concept of center of mass and be able to answer questions regarding how the center of mass moves during collisions. I tend to use the term “system” when talking to my own students, but formally it is the center of mass of the system. The important thing to understand is that you have been treating all objects as point masses located at the center of mass already. Now we just need to apply the idea of a point mass to a two object system undergoing a collision.
If you remember one thing about this episode, remember that as long as there are no external forces acting on a system, then (because momentum is conserved) the momentum of the center of mass is constant. If the mass of the system does not change, then the velocity must also be constant in order to adhere to the law of conservation of momentum. Let’s try an example.
Two ice hockey players of equal mass (100 kg) are 1 m apart. One player remains at rest while the other coasts towards the stationary player with a velocity of 0.5 m/s. After 1 second, the players are 0.5 m apart. What is the velocity of the center of mass for the two player system? Well, because they are equal in mass, the center of mass is always exactly halfway from one player to the other. So, the center of mass moved from 0.5 m to 0.25 meters in one second. That’s a velocity of the center of mass of 0.25 m/s.
The two players then collide through an inelastic collision, stick together and continue along the ice with a velocity of 0.25 m/s. Don’t believe me? Check using conservation of momentum. You will see that the player in motion has a momentum of 50 kg*m/s while the other has a momentum of 0.0 kg*m/s. The system, or center of mass has a momentum of 50 kg*m/s then. Since there are no external forces mentioned, momentum is conserved and the two 100 kg players, 200 kg total, move together with a velocity of 0.25 kg*m/s to maintain the 50 kg*m/s momentum.
To Recap…
The center of mass is a point in space where the weighted average mass of a system lies. The center of mass’s velocity and momentum are constant in the absence of a net external force acting on the system. Some say the momentum of the center of mass is conserved, others say the momentum of the system is conserved. Tomato, Tomato.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at the simple pendulum, and the factors that affect how quickly it oscillates.
Today’s Question of the Day focuses on the center of mass of a system.
Question: Two friends attempt to balance on the ends of a seesaw of length L. One friend is 3 times as massive. If the heavier friend stands on one end of the board, where should the fulcrum be moved to be under the center of mass?