Seeking intuition on movement of COM

In summary: This means that the center of mass will not move horizontally, even as the box slides down the ramp. This is because the forces exerted by gravity and the normal force are always equal and opposite, canceling each other out. Therefore, the center of mass formula is based on the fact that the horizontal component of the center of mass will remain constant in this scenario. This is derived from the principles of conservation of energy and momentum. In summary, the center of mass formula in this scenario is based on the conservation of energy and momentum, as well as the fact that the horizontal component of the center of mass will remain constant due to equal and opposite forces canceling each other out.
  • #1
fahraynk
186
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Basically a weight moves from starting point on a ramp down to the bottom of the ramp. The ramp is on wheels so the ramp will also move. The formula is based on the center of mass of the combined object. Center of mass does not move.
XA0 = position of small mass before it slides down ramp
XB0 = position of COM of ramp before box slides
XA,XB= new position after box slides

$$X_{A0}M1+X_{B0}M2 = X_{COM}(M1+M2)\\
X_{A}M1+X_{B}M2 = X_{COM}(M1+M2)\\
X_{A0}M1+X_{B0}M2 = X_{A}M1+X_{B}M2$$

My question is where does this come from? Assuming I did not know this through repetition that the COM would not move... what physical laws would you apply to come to this conclusion?

If the box is sitting on the ramp, there should be a force normal to the ramp as M1gSin(theta). Isn't this the force on the ramp whether the mass is moving or not? If the ramp is not moving before the box moves that means friction between the ramp and floor is holding it in place. So... that will still be canceled out whether the box is moving or not. Where does this center of mass formula come from??
 
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  • #2
Conservation of energy.
Conservation of momentum.
 
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  • #3
You can do a free body diagram. It's not a closed system because you have gravity pulling down, and the ramp is being supported by the ground which provides an upward normal force. But you can see that gravity and the normal force are completely vertical. There is no horizontal component to the external force on the system, so there is no change in the horizontal component of the center of mass.
 

What is the center of mass (COM) and why is it important in movement?

The center of mass (COM) is the point in an object or system where all of its mass can be considered to be concentrated. It is important in movement because it determines the overall stability and balance of the object or system. In humans, the COM is located in the pelvis and plays a crucial role in maintaining an upright posture and coordinating movements.

How can we measure and calculate the movement of COM?

The movement of COM can be measured and calculated using motion capture technology, which tracks the movement of reflective markers placed on the body. These markers are then used to calculate the position, velocity, and acceleration of the COM. Other methods include force plates, which measure the forces acting on the body, and biomechanical modeling, which uses mathematical equations to estimate the movement of COM.

What factors affect the movement of COM?

The movement of COM can be affected by several factors, including the position and size of body segments, muscle activity, external forces, and the environment. Changes in any of these factors can alter the movement of COM and affect the overall stability and balance of the body.

How does understanding the movement of COM benefit sports and exercise?

Understanding the movement of COM is crucial for improving performance and reducing the risk of injury in sports and exercise. By analyzing the movement of COM, coaches and trainers can identify areas for improvement and develop training programs to optimize movement efficiency and stability. In addition, understanding the movement of COM can also help prevent injuries by identifying potential risks and implementing strategies to reduce their impact.

What are some real-world applications of studying the movement of COM?

The study of COM has many real-world applications, including in the fields of biomechanics, physical therapy, and robotics. In biomechanics, understanding the movement of COM is essential for designing efficient and safe movements in sports and other activities. In physical therapy, it is used to assess and improve balance and gait in patients with movement disorders. In robotics, the concept of COM is used to design more stable and efficient movements in humanoid robots.

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