Center of Mass of acceleration and velocity

In summary, the conversation discusses finding the acceleration and velocity of a two-body system, specifically the center of mass. The formula for the acceleration of the center of mass is a weighted average of the individual masses and accelerations. To find the coordinates of the center of mass, the coordinates of each mass can be written in (x,y) form and then the weighted average can be taken. Differentiation is necessary to find the velocity and acceleration of the center of mass.
  • #1
Riman643
62
2
Homework Statement
The figure shows an arrangement with an air track, in which a cart is connected by a cord to a hanging block. The cart has mass m1 = 0.670 kg, and its center is initially at xy coordinates (–0.550 m, 0 m); the block has mass m2 = 0.290 kg, and its center is initially at xy coordinates (0, –0.280 m). The mass of the cord and pulley are negligible. The cart is released from rest, and both cart and block move until the cart hits the pulley. The friction between the cart and the air track and between the pulley and its axle is negligible. (a) In unit-vector notation, what is the acceleration of the center of mass of the cart–block system? (b) What is the velocity of the com as a function of time t, in unit-vector notation?
Relevant Equations
F = ma
com equation
I don't understand what I am supposed to do. I know how to find the acceleration of the system, but I am not sure how to find the com acceleration. My guess would be something along the lines of xcom = (m1a1*x + m2a2*x)/(m1a1 +m2a2). Then do the same for the y axis. Then to find the velocity all I would have to do would be to find the Integral of com of Acceleration? I am stuck.
 

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  • #2
Riman643 said:
something along the lines of xcom = (m1a1*x + m2a2*x)/(m1a1 +m2a2)
Somewhat like that, yes.
But you want an answer which is an acceleration vector, so you don't want to be dividing by accelerations.
Try considering one coordinate at a time.
 
  • #3
haruspex said:
Somewhat like that, yes.
But you want an answer which is an acceleration vector, so you don't want to be dividing by accelerations.
Try considering one coordinate at a time.
Well I know in either direction, one of the forces will be canceled out because the each mass as a zero in their coordinates. Would I want instead xcom = (m1a1*x)/(m1+m2)?
 
  • #4
You are not stuck. The real formula for the acceleration of the center of mass is much simpler and involves only m1, m2 and a1 and a2. Integrating that will be much simpler.
To compute the postition, the speed or the accelaration of the center mass you use the same formula each time, based on calculating a weighted average.
 
  • #5
willem2 said:
You are not stuck. The real formula for the acceleration of the center of mass is much simpler and involves only m1, m2 and a1 and a2. Integrating that will be much simpler.
To compute the postition, the speed or the accelaration of the center mass you use the same formula each time, based on calculating a weighted average.
So are the x and y coordinates useless then in this equation? They are throwing me off.
 
  • #6
Look at your lecture notes or books. How is the position of the center of mass defined?
Velocity is the time derivative of position; acceleration is time derivative of the velocity.
So what are the components of the acceleration vector of a two-body system in terms of the individual accelerations?
 
  • #7
It is weighted average of mass and position. I just don't know how to apply it to this problem. The individual accelerations of this equation are 2.96 and -2.96. I just do not know how to convert this to a center of mass equation.
 
  • #8
Riman643 said:
It is weighted average of mass and position. I just don't know how to apply it to this problem. The individual accelerations of this equation are 2.96 and -2.96. I just do not know how to convert this to a center of mass equation.
Write the coordinates of each mass in the (x,y) form and take the weighted average to find the coordinates of the mass centre. Then differentiate as necessary.
 
  • #9
haruspex said:
Write the coordinates of each mass in the (x,y) form and take the weighted average to find the coordinates of the mass centre. Then differentiate as necessary.
I'm still confused. Differentiate how? I found the center of mass but how do I differentiate it to give me velocity and acceleration?
 
  • #10
So I found the correct answer to the acceleration problem. How do I integrate the vector notation of the acceleration to give me the velocity?
 
  • #11
Never mind. I figured it out. I'm stupid.
 

Related to Center of Mass of acceleration and velocity

1. What is the center of mass of acceleration and velocity?

The center of mass of acceleration and velocity is the point at which the mass of an object is evenly distributed and the object experiences no net external force. It is also known as the center of gravity.

2. How is the center of mass of acceleration and velocity calculated?

The center of mass can be calculated by taking the weighted average of the positions of all the individual particles in a system. It can also be calculated using the formula xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where xcm is the center of mass and mn and xn are the mass and position of each particle.

3. Why is the center of mass of acceleration and velocity important?

The center of mass is important because it helps us understand the motion and stability of objects. It is also a useful concept in physics, as it allows us to simplify complex systems and make predictions about their behavior.

4. How does the center of mass of acceleration and velocity affect an object's motion?

The center of mass affects an object's motion by determining how the object will respond to external forces. If the center of mass is located at the geometric center of an object, the object will experience no rotational motion when subjected to a force. However, if the center of mass is not at the geometric center, the object will experience both translational and rotational motion.

5. Can the center of mass of acceleration and velocity change?

Yes, the center of mass can change if the distribution of mass within an object changes. For example, if an object loses mass on one side, the center of mass will shift towards the remaining mass. Additionally, external forces can also cause the center of mass to change, as they can alter the velocity and acceleration of individual particles in the system.

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