System of Particles and Rotational Motion · Lesson 4 of 15
Motion of the Centre of Mass and Linear Momentum
“Derive the external-force law for a system and separate overall motion from internal motion.”
• Derive CM velocity and acceleration. • Explain cancellation of internal forces. • Apply momentum conservation to closed constant-mass systems. • Separate translational and internal kinetic energy.
Differentiating the mass-weighted position
Choose a system containing the same particles throughout the calculation and assume their masses stay constant. Starting with MR = Σmᵢrᵢ, differentiate once with time. Since masses are constants, MV = Σmᵢvᵢ. Differentiating again gives MA = Σmᵢaᵢ. Each mᵢaᵢ is the total force on that particle, including internal forces from other system particles and external forces from outside.
The CM moves as if the total mass were concentrated there and acted on by the resultant external force. This statement determines translational motion, not the complete rotation or deformation. The internal interactions can be complicated, but need not be known to calculate CM acceleration. The argument uses Newton’s second law for particles and Newton’s third law for the internal pairs.
What remains unchanged without an external force?
If the resultant external force is zero, total momentum and CM velocity remain constant. Constant may mean zero or a nonzero vector. Conservation can also apply to one component: on a smooth horizontal floor, horizontal momentum can be conserved even while gravity and support forces act vertically. Identify the chosen system and direction before invoking the law.
Problem
A child and trolley initially move together at speed V on a smooth horizontal floor. The child then runs around. What happens to their CM velocity?
- 1.Take child plus trolley as the system. The child–trolley forces are internal, and there is no horizontal external force.
- 2.The combined CM keeps its initial horizontal velocity V. Child and trolley velocities can each change while their mass-weighted average stays fixed.
- 3.If an external horizontal force is introduced, such as pushing against a wall, the conclusion must be reconsidered.
A projectile exploding in a uniform gravitational field provides another example. The explosion forces are internal; gravity still gives the combined CM downward acceleration g. Its path continues along the parabola of the unbroken projectile, provided all fragments are included and air resistance is negligible. Individual fragments follow different paths. Momentum of the whole system is not constant during the complete flight because gravity is external.
In an ideal decay observed in the CM frame, two products have equal and opposite momenta, so they move back-to-back. In a laboratory frame with moving CM the paths look different. Binary stars similarly combine CM motion with orbital motion about it. Circular equal-mass orbits are a simple illustration; real binary orbits need not be circular. Choose the CM frame to make the internal pattern easier to interpret.
Separating the energy
Substitute vᵢ = V + vᵢ′ into the particle energy sum and expand the square. The first contribution is MV²/2. The cross term is V · Σmᵢvᵢ′, which vanishes because total momentum relative to the CM is zero. The remaining sum is K′. For a rigid body, internal relative motion can be rotation; for a general system it may include many other motions. Conservation of momentum does not require K′ to stay fixed during an explosion or an inelastic interaction.
Quiz
What controls acceleration of a system’s CM?
MA equals the vector sum of external forces.
Zero external force implies CM velocity is what?
A constant nonzero velocity is allowed.
During an explosion in uniform gravity, which follows the original projectile parabola?
Internal explosion forces do not change CM acceleration.
In the CM frame, total linear momentum is what?
The mass-weighted relative velocities sum to zero.
In K = MV²/2 + K′, K′ describes what?
It is the kinetic energy of the relative particle motions.
Key Takeaways
• P = MV connects particle momentum with CM motion. • Internal forces cancel in the system force sum. • Zero external force conserves momentum, not necessarily kinetic energy. • Separate CM translation from motion relative to CM.