# Solving for the trajectory of the center of mass

• I
• nDever
In summary, the conversation discusses the physics of a square with mass M that is free to translate and rotate in the xy plane. An angular impulse is generated when a linear impulse is applied at a point above the center of mass, causing both rotational and translational motion. The overall motion of the system can be computed using momentum and angular momentum conservation.
nDever
I'm working on the physics engine component of a game engine I'm building, and I need some guidance with this particular situation.

Consider a square with mass M that is free to translate in the xy plane and free to rotate about any axis perpendicular to the page (Fig. 1)

If a linear impulse J is applied at a point above the center of mass (CM) as shown below, I know there must be some angular impulse (momentary torque) generated since there is a component of J that is perpendicular to the displacement vector from CM. I imagine this angular impulse will tend to rotate the square clockwise.

However, I can also imagine that the CM will also undergo translation since the square is not constrained. How would I go about computing the overall rotational + translational motion of this system?

Delta2
The linear motion is the easiest since overall momentum conservation gives
$$\frac{d\vec p}{dt} = m\dot{\vec v}_{com} = \vec F_{tot}$$
The rotational part can be slightly trickier due to the overall acceleration of the system. You should be able to do it by overall angular momentum, preferably relative to the CoM in the comoving frame.

vanhees71

## 1. What is the center of mass?

The center of mass is the point in an object or system where the mass is evenly distributed in all directions. It is also known as the center of gravity.

## 2. Why is it important to solve for the trajectory of the center of mass?

Solving for the trajectory of the center of mass allows us to predict the motion and behavior of an object or system. This information is crucial in understanding and analyzing physical phenomena.

## 3. How is the trajectory of the center of mass calculated?

The trajectory of the center of mass is calculated using the principles of Newtonian mechanics, specifically the laws of motion and the concept of the center of mass. It involves determining the position, velocity, and acceleration of the center of mass at different points in time.

## 4. What factors affect the trajectory of the center of mass?

The trajectory of the center of mass is affected by various factors such as the mass distribution of the object or system, external forces acting on the object, and the initial conditions of the object's motion.

## 5. Can the trajectory of the center of mass be calculated for any object or system?

Yes, the trajectory of the center of mass can be calculated for any object or system as long as the principles of Newtonian mechanics apply. However, for complex systems, the calculations may become more challenging and may require the use of advanced mathematical techniques.

• Mechanics
Replies
5
Views
1K
• Mechanics
Replies
24
Views
2K
• Mechanics
Replies
11
Views
1K
• Mechanics
Replies
5
Views
2K
• Mechanics
Replies
2
Views
1K
• Mechanics
Replies
14
Views
2K
• Mechanics
Replies
19
Views
3K
• Mechanics
Replies
2
Views
1K
• Mechanics
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
17
Views
919