Degrees of Freedom for Rigid Body of n Particles

In summary, the degree of freedom for a rigid body made up of n particles is 3n-nC2. This means that for n particles, the degree of freedom is 3n, but if there are 2 particles with a fixed distance between them, the degree of freedom is n 3(2)-1=5. For three particles, the degree of freedom is 7. However, even if the distance between particles is fixed, the degree of freedom is still 3n-(n-1), which is equivalent to 3n-nC2. The term C2 represents the number of combinations, and in this case, it refers to the combination of 2 particles. In general, a rigid body has
  • #1
ajayguhan
153
1
How is for rigid body made up of n particle, the degree of freedom is 3n-nC2.i can understand that degree freedom of n particle is 3n and suppose you have 2 particle whose distance between them is fixed, then degree of freedom is n 3(2)-1=5.if we have three particles then it is 7.
Therefore n particle whose distance between them is fixed have degree of freedom which is 3n-(n-1) .how can it be 3n-.nC2.
 
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  • #2
What is C2?
 
  • #3
It does not matter how many particles a rigid body is made of (except when all of them are collinear). It always has 6 degrees of freedom. 3 rotational (Euler angles), and 3 translational. All-collinear rigid bodies, even though the term "body" is hardly applicable here, lose one rotational DoF.
 

FAQ: Degrees of Freedom for Rigid Body of n Particles

1. What is the definition of "degrees of freedom" for a rigid body of n particles?

Degrees of freedom refer to the number of independent variables required to fully describe the motion of a rigid body consisting of n particles. It represents the number of ways in which the body can move without violating any constraints.

2. How is the degrees of freedom calculated for a rigid body of n particles?

The degrees of freedom can be calculated using the formula 6n - k, where n is the number of particles in the body and k is the number of constraints or equations of motion. This formula is derived from the fact that each particle has 6 possible degrees of freedom (3 translational and 3 rotational) and the constraints reduce the total number of degrees of freedom.

3. Can a rigid body of n particles have a negative degrees of freedom?

No, a rigid body of n particles cannot have a negative degrees of freedom. The minimum number of degrees of freedom for a rigid body is 0, which means that the body is completely constrained and cannot move at all.

4. How does the number of degrees of freedom affect the complexity of a rigid body's motion?

The higher the number of degrees of freedom, the more complex the motion of a rigid body becomes. This is because a higher number of degrees of freedom allows for more diverse and intricate movements of the body, making it more difficult to predict and analyze its motion.

5. Can the degrees of freedom for a rigid body of n particles change?

Yes, the degrees of freedom for a rigid body of n particles can change depending on the constraints applied to the body. For example, if a new constraint is introduced, it may reduce the degrees of freedom and limit the body's motion in a certain direction.

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