Equation of motion for a rigid body

In summary: That is correct.In summary, the conversation discussed the equation of motion for an arbitrary rigid body with a force acting on it at a point A, which is not the center of mass C. The idea was to separate out translation and rotation, with the whole body moving by F=ma and rotation given by D=r x F where r is the distance between A and C. However, there was some uncertainty about this approach. The conversation also mentioned the relationship between torque and angular momentum, and the additional term that arises when operating in a rotating frame. Finally, it was confirmed that F=ma is a valid equation for translation, regardless of where the force is applied on the body.
  • #1
Gavroy
235
0
hi

let me says you have an arbitrary rigid body and a force F acts on this body at some point A, which is not the center of mass that is called C.

how do you get the equation of motion?

my idea was to separate out translation and rotation:

maybe the whole body moves by F=ma
and the rotation is given by: D=r x F where r is the distance between A and C.

but i am not sure at all about this.

by the way: i am rather looking for a general idea that would give me the equation of motion.
 
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  • #2
Hi Gavroy! :smile:
Gavroy said:
my idea was to separate out translation and rotation:

maybe the whole body moves by F=ma
and the rotation is given by: D=r x F where r is the distance between A and C.

yes, F = ma,

and r x F = Iα

where I is the moment of inertia about an axis through the centre of mass and parallel to r x F

(but if that axis is not a principal axis of the body, I needs to be the inertia tensor)
 
  • #3
tiny-tim said:
r x F = Iα
That isn't quite correct. What is valid is that angular momentum is the product of the inertia tensor and angular velocity: [itex]\vec L = \mathbf I\,\vec{\omega}[/itex].

Now differentiate both sides. The left hand side is easy: It's torque, which is given by [itex]\vec {\tau} = \vec r \times \vec F[/itex]. The right hand side is a bit tougher. In which frame? Which derivative? The inertia tensor is time varying from the perspective of an inertial frame. The inertia tensor for a rigid body is constant in a frame fixed with respect to the body, but now you have the problem of take time derivatives in a non-inertial frame. It's easier than making the inertia tensor time varying, but it is not that simple r x F = Iα. Instead you get
[tex]\vec r \times \vec F = \mathbf I \frac{d\vec{\omega}}{dt} + \vec{\omega}\times (\mathbf I\,\vec{\omega})[/tex]
 
  • #4
D H said:
[tex]\vec r \times \vec F = \mathbf I \frac{d\vec{\omega}}{dt} + \vec{\omega}\times (\mathbf I\,\vec{\omega})[/tex]

interesting, thank you. can you give me a hint where this extra term comes from or how i can get this one? (i am referring to the cross prouct (angular velocity cross inertia tensor dot angular velocity)

by the way: where i right about the translation given by F=ma? i am a little sceptical now, because if this equation is right it would not make a difference (referring to the translation) whether the force acts on the center of mass or an arbitrarily chosen other point of the body?
 
  • #5
That extra term results from operating in a rotating frame. For any vector quantity [itex]\vec q[/itex], the time derivatives of that vector from the perspectives of inertial and rotating observers are related via
[tex]
\left(\frac {d\vec q}{dt}\right)_{\text{inertial}} =
\left(\frac {d\vec q}{dt}\right)_{\text{rotating}} +\quad
\vec{\omega} \times \vec q[/tex]
This is called the transport theorem. Wiki reference: http://en.wikipedia.org/wiki/Rotati...tives_in_the_two_frames[url]. And yes, F=ma.
 

1. What is the equation of motion for a rigid body?

The equation of motion for a rigid body is a mathematical representation of the motion of a body in space. It describes how the position, velocity, and acceleration of a rigid body change over time.

2. How is the equation of motion for a rigid body derived?

The equation of motion for a rigid body is derived from the principles of classical mechanics, specifically Newton's second law of motion which states that the net force acting on an object is equal to the product of its mass and acceleration.

3. What are the variables in the equation of motion for a rigid body?

The variables in the equation of motion for a rigid body include the position, velocity, and acceleration of the body, as well as the mass and the forces acting on the body.

4. What is the significance of the equation of motion for a rigid body?

The equation of motion for a rigid body allows us to predict and analyze the motion of a body in space. It is used in various fields of science and engineering, such as physics, mechanics, and robotics, to understand and control the motion of objects.

5. Are there any limitations to the equation of motion for a rigid body?

While the equation of motion for a rigid body is a powerful tool for understanding and predicting motion, it does have limitations. It assumes that the body is rigid, meaning that it does not deform or change shape during motion. It also does not take into account factors such as air resistance and friction, which can affect the motion of real-world objects.

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