Show that when the total external force acting on N particles is zero, the total external torque is independent of the choice of reference point O.
I will do it in two dimensions, similar in 3
moment = F1 X R1 + (-F1) X R2
= (R1y Fx - R1x Fy)k -(R2y Fx -R2x Fy)k
= Fx(R1y-R2y)k +Fy (R2x-R1y)k
so in k direction and depending only on how far apart the two forces are, not the origin of R1 and R2
typo
= Fx(R1y-R2y)k + Fy (R2x-R1x)k
By the way
the vector product
F X (R2-R1) is a moment called a "couple".
To show that when the total external force acting on N particles is zero, the total external torque is independent of the choice of reference point O, we can use the definition of torque and the concept of the center of mass.
1. Definition of Torque:
The torque τ about a reference point O is given by the cross product of the position vector r and the force vector F, mathematically expressed as: τ = r x F, where x represents the cross product.
2. Center of Mass:
The center of mass of a system of particles is a point that represents the average position of the particles' masses. For a system of N particles, the center of mass is calculated as: R = (m1r1 + m2r2 + ... + mNrN) / (m1 + m2 + ... + mN), where mi represents the mass of particle i and ri represents the position vector of particle i from the reference point O.
Now, let's proceed with the proof:
Step 1: Assume that the total external force acting on the N particles is zero. Mathematically, this can be expressed as: ∑F_ext = 0, where ∑F_ext represents the summation of external forces.
Step 2: Take the torque about the reference point O due to each individual particle. For each particle i, τi = ri x Fi.
Step 3: Sum all the individual torques to obtain the total external torque. Mathematically, this can be expressed as: ∑τ_ext = ∑(ri x Fi).
Step 4: Apply the cross product distributive property: ∑(ri x Fi) = (∑ri) x (∑Fi).
Step 5: Express the center of mass as R = (∑(miri)) / (∑mi).
Step 6: Rewrite the total external torque using the center of mass: ∑τ_ext = (∑ri) x (∑Fi) = R x (∑Fi).
Step 7: Since the total external force acting on the N particles is assumed to be zero (∑F_ext = 0), we have ∑Fi = 0.
Step 8: Consequently, ∑τ_ext = R x 0 = 0.
Conclusion: We have shown that when the total external force acting on N particles is zero, the total external torque ∑τ_ext is also zero. This implies that the choice of reference point O does not affect the total external torque, as it always equals zero.