The Angular Momentum of a System of Particles — Question 6

Question 6

Uniform gravity accelerates the center without changing internal angular momentum

A collection of particles interacts through internal central forces and moves in uniform gravity. Ignore air resistance. Define internal angular momentum using positions and velocities relative to its center of mass.

1. Find the external gravity torque about the center.

2. Explain whether internal angular momentum changes even though the center accelerates.

Question 6 - Solution

Uniform gravity accelerates the center without changing internal angular momentum

(a) Derivation and calculation

About the center, the gravity torque is $$\vec\tau_C=\sum\vec\rho_i\times m_i\vec g =\left(\sum m_i\vec\rho_i\right)\times\vec g.$$ By the center-of-mass identity, $$\boxed{\vec\tau_C=\vec0}.$$

(b) Result and interpretation

For relative velocities, differentiating internal angular momentum introduces the acceleration correction $$-\sum\vec\rho_i\times m_i\vec a_C =-\left(\sum m_i\vec\rho_i\right)\times\vec a_C=0.$$ Internal central-force torques cancel, so $$\boxed{\frac{d\vec L_C}{dt}=\vec\tau_C=0},\qquad\vec a_C=\vec g.$$ The center can accelerate while internal angular momentum stays constant.

Physical check

About another fixed origin, gravity can have a nonzero moment and change the orbital contribution.