# Center of Mass Motion Question

• Oriako
In summary, the person at the left end of the beam moved due to the ball being thrown from person to person. The new x-position of the person is at (3, 0). The ball must be walk back to the other person in order for them to be at the same x-position.
Oriako

## Homework Statement

A person with mass m1 = 69 kg stands at the left end of a uniform beam with mass m2 = 108 kg and a length L = 2.5 m. Another person with mass m3 = 67 kg stands on the far right end of the beam and holds a medicine ball with mass m4 = 9 kg (assume that the medicine ball is at the far right end of the beam as well). Let the origin of our coordinate system be the left end of the original position of the beam as shown in the drawing. Assume there is no friction between the beam and floor.

3)What is the new x-position of the person at the left end of the beam? (How far did the beam move when the ball was throw from person to person?

4)To return the medicine ball to the other person, both people walk to the center of the beam. At what x-position do they end up?

## Homework Equations

$$\frac{1}{M_{total}} \sum_{i=1}^{n} m_{i}r_{i}$$

## The Attempt at a Solution

I answered the first 2 parts of the question that have to do with finding the center of mass and I am fine with any sort of easy "find the center of mass of this situation", but I cannot figure out these sort of relative motion questions to do with the distance something is moving from the center of mass.

I understand that since there are no external forces, the center of mass remains at the same place, I just don't know how to quantify what the motion of ball being thrown from one side to the other would do and I have trouble with relative motion questions.

The question is just for practice and I want to figure it out. Any help would be greatly appreciated.

Don't worry about the motion. All that matters is the change in location of the ball in 3) and of the people and the ball in 4).

For 3), determine the location of the centre of mass of the beam + people + ball in the frame of reference of the beam before and after the ball changes position. You know that the location of the centre of mass relative to the floor does not change, so that will tell you how far the beam has to move due to the change in position of the ball.

Do the same for 4).

AM

Yes, I've got that far but how is it that "it tells me how far the beam has to move due to the change in the position of the ball" I conceptually understand everything you mentioned and I've calculated everything I just can't put it together.

So I've finally figured out question 3 but now I'm lost on 4. Any ideas?

Oriako said:
So I've finally figured out question 3 but now I'm lost on 4. Any ideas?
If everyone moves to the centre of the beam, where is the centre of mass of the system (beam + people + ball) in the frame of reference of the beam? Where was the centre of mass of the system before in the beam frame of reference? Does the centre of mass of the system move in the reference frame of the floor? So how far does the beam move relative to the floor so that the cm of the system is in the same place relative to the floor?

AM

## 1. What is the center of mass?

The center of mass is the point at which an object can be balanced perfectly without any external forces acting on it. It is often referred to as the "center of gravity" or "centroid."

## 2. How is the center of mass calculated?

The center of mass can be calculated by finding the weighted average of an object's individual masses and their respective distances from a reference point. This can be done using the equation:
Center of mass = (m1x1 + m2x2 + m3x3 + ... + mnxn) / (m1 + m2 + m3 + ... + mn)

## 3. Why is the center of mass important?

The center of mass is important because it helps us understand how an object will move and behave under the influence of external forces. It is also useful in determining an object's stability and how it will respond to different types of motion.

## 4. Can the center of mass be outside of an object?

Yes, the center of mass can be outside of an object. This is often the case for irregularly shaped objects or objects with asymmetrical mass distributions.

## 5. How does the center of mass relate to rotational motion?

The center of mass is important in rotational motion because it is the point around which an object will rotate. The distribution of mass around the center of mass determines an object's moment of inertia, which affects its rotational motion.

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