Energy Methods for Particle Systems

Energy Methods for Particle Systems

The work-energy theorem for one particle extends directly to a system of particles, but the extension introduces an important distinction between external and internal interactions.

For a system containing particles i = 1,…,N, define the total kinetic energy

(1)

Applying the work-energy theorem to every particle and adding the results gives

(2)

where W_ext is the actual work done by external forces on the particles and W_int is the actual work done by internal forces.

The total kinetic energy can also be decomposed into motion of the center of mass and motion relative to the center of mass:

(3)

These two relations form the foundation of energy methods for particle systems.

1 Total kinetic energy of a particle system

For particles with masses m_i and velocities v_i,

(4)

Kinetic energy is additive. Each particle contributes

(5)

The total mass is

(6)

The center-of-mass position is

(7)

Differentiating,

(8)

Therefore the total linear momentum is

(9)

2 System work-energy theorem

For particle i,

(10)

Write the force on particle i as the sum of external and internal forces:

(11)

Then

(12)

Sum over all particles:

(13)

Since

(14)

define

(15)

and

(16)

Thus

(17)

This is the work-energy theorem for a particle system.

3 Internal forces cancel in momentum, but not generally in work

Newton’s third law causes internal forces to cancel from the net force on the system when the internal forces occur in equal-and-opposite pairs:

(18)

That cancellation gives

(19)

It does not imply that internal work vanishes.

For a pair of particles i and j,

dWijint	= F ij ⋅ dri + Fji ⋅ drj	(20)
	= Fij ⋅ (dri − drj).	(21)

Therefore

(22)

This is generally not zero because the two particles need not undergo the same displacement.

4 Conservative internal forces and internal potential energy

Suppose the internal force between each interacting pair is conservative.

For pair i,j, define a pair potential

(23)

Then the internal pair work satisfies

(24)

Summing over distinct pairs gives the internal potential energy

(25)

Hence

(26)

for conservative internal interactions.

The system work-energy theorem becomes

(27)

Therefore

(28)

If external conservative potentials are also included in U, then the same bookkeeping extends to the full mechanical energy.

5 Center-of-mass and relative velocities

Define the velocity of each particle relative to the center of mass:

(29)

Therefore

(30)

The relative velocities satisfy

(31)

To verify this,

∑ imivi′	= ∑ imi(vi − VCM)	(32)
	= ∑ imivi − VCM ∑ imi	(33)
	= MVCM − MVCM	(34)
	= 0.	(35)

6 Derivation of the kinetic-energy decomposition

Start with

(36)

Expand the square:

K

= ∑ imiV CM2 + ∑ imiVCM ⋅ vi′ + ∑ imi 2.

(37)

The first term is

(38)

The cross term is

∑ imiVCM ⋅ vi′	= VCM ⋅∑ imivi′	(39)
	= 0.	(40)

Therefore

(41)

Define

(42)

and

(43)

Thus

(44)

7 Physical meaning of the decomposition

The term

(45)

is the kinetic energy the system would have if all its mass moved together with the center of mass.

The term

(46)

measures kinetic energy associated with motion of the particles relative to the center of mass.

Relative kinetic energy can include

• particles approaching or separating,
• vibration,
• orbital motion about the center of mass,
• rotation of a rigid particle assembly,
• random microscopic motion in a many-particle system.

The exact interpretation depends on the physical model.

8 Example 1: two-particle kinetic-energy decomposition

Two particles move along the x axis.

Let

m1	= 2.0 kg,	v1	= 4.0 m∕s,	(47)
m2	= 3.0 kg,	v2	= −1.0 m∕s.	(48)

The total mass is

(49)

The center-of-mass velocity is

V CM	=	(50)
	=	(51)
	= 1.0 m∕s.	(52)

The total kinetic energy is

K	= (2)(42) + (3)(12)	(53)
	= 17.5 J.	(54)

The center-of-mass kinetic energy is

(55)

The relative velocities are

v1′	= 4 − 1 = 3 m∕s,	(56)
v2′	= −1 − 1 = −2 m∕s.	(57)

Thus

Krel	= (2)(32) + (3)(22)	(58)
	= 15 J.	(59)

Therefore

(60)

9 Center-of-mass equation and pseudowork

The center of mass obeys

(61)

Dot this equation with

(62)

Then

(63)

Integrating gives

(64)

The integral on the left is often called pseudowork.

It must not be confused with the actual external work

(65)

Actual work uses the displacement of each force’s point of application. Pseudowork uses the displacement of the center of mass.

They are generally different.

10 Example 2: actual work versus pseudowork

Consider two equal particles of mass m initially at rest. Particle 2 remains fixed while a constant external force F acts on particle 1 and moves it through distance d.

The actual external work is

(66)

The center of mass moves only half as far:

(67)

The net external force on the system is

(68)

Therefore the pseudowork is

(69)

The pseudowork equals the change in center-of-mass kinetic energy:

(70)

The actual external work equals the change in total kinetic energy:

(71)

The difference appears as relative kinetic energy:

(72)

11 Why the distinction matters

For a single particle,

(73)

so actual work and pseudowork coincide.

For a multi-particle system, they generally do not.

The difference between external work and center-of-mass pseudowork can change

• relative kinetic energy,
• rotational kinetic energy,
• vibrational energy,
• elastic energy,
• internal energy.

Therefore one should not write

(74)

unless the conditions of the specific problem justify it.

12 Two-particle relative motion and reduced mass

For two particles, define the relative coordinate

(75)

and relative velocity

(76)

Define the reduced mass

(77)

For a two-particle system,

(78)

Therefore

(79)

This decomposition is central to two-body mechanics, collisions, binary orbits, and molecular motion.

13 Derivation of the two-particle relative kinetic energy

The center-of-mass condition is

(80)

The relative velocity is

(81)

Solving,

v1′	= vrel,	(82)
v2′	= −vrel.	(83)

Then

Krel	= m1 2 + m2 2	(84)
	= vrel2	(85)
	= vrel2	(86)
	= vrel2.	(87)

Therefore

(88)

14 Internal springs

Consider two particles connected by an ideal spring.

The spring interaction is internal to the two-particle system.

Let the spring extension from its natural length be

(89)

The internal potential energy is

(90)

If no external work is done,

(91)

Using the center-of-mass decomposition,

(92)

If no net external force acts,

(93)

is separately constant, so the spring exchanges energy with the relative motion.

15 Explosions and release of internal energy

Suppose a system is initially at rest and an internal process causes it to separate into pieces.

If the net external impulse is negligible,

(94)

remains constant.

If the initial center of mass is at rest,

(95)

before and after the event.

Therefore

(96)

Yet the fragments can acquire substantial kinetic energy.

That kinetic energy is relative kinetic energy and must come from another internal energy reservoir, such as

• chemical energy,
• elastic energy,
• nuclear energy,
• stored pressure energy.

16 Example 3: two-fragment explosion

A system initially at rest separates into two fragments:

m1	= 2.0 kg,	(97)
m2	= 3.0 kg.	(98)

After separation, fragment 1 moves at

(99)

Momentum conservation gives

(100)

Thus

v2	= −v1	(101)
	= −(6)	(102)
	= −4.0 m∕s.	(103)

The final kinetic energy is

Kf	= (2)(62) + (3)(42)	(104)
	= 36 + 24	(105)
	= 60 J.	(106)

Because the center of mass remains at rest,

(107)

Therefore all 60 J is relative kinetic energy:

(108)

At least 60 J of internal energy has been converted into kinetic energy in this idealized model.

17 External conservative potentials for a system

Suppose each particle also moves in an external conservative field.

For example, near Earth’s surface,

(109)

If The Gravitational Field is uniform,

Ug	= g ∑ imiyi	(110)
	= MgY CM.	(111)

Therefore

(112)

In a uniform gravitational field, the gravitational potential energy of the entire particle system depends only on the center-of-mass height.

This is a useful simplification.

18 General mechanical-energy accounting for a particle system

Let

(113)

include all conservative interactions represented by potentials.

Then define

(114)

Any remaining nonconservative external work or modeled dissipative transfer changes this mechanical energy:

(115)

Using the kinetic-energy decomposition,

(116)

This form makes clear where energy can reside within a particle system.

19 Internal nonconservative interactions

Not all internal interactions are conservative.

A system can contain

• internal friction,
• inelastic deformation,
• damping,
• collisions with permanent deformation,
• microscopic dissipation.

In such cases, macroscopic mechanical energy can decrease even if no energy crosses the system boundary.

The missing macroscopic mechanical energy appears as internal energy.

For a closed system,

(117)

in a simplified model with no other energy transfer.

Thus the larger total-energy balance remains valid even when the mechanical-energy balance does not close by itself.

20 A system-energy workflow

For a multi-particle energy problem:

1. Define the particles included in the system.
2. Identify the center-of-mass motion.
3. Decide whether relative motion matters.
4. Separate external and internal interactions.
5. Represent conservative interactions with potential-energy functions.
6. Identify nonconservative work or internal-energy conversion.
7. Use

   (118)

8. Apply the appropriate work-energy or mechanical-energy equation.
9. Check momentum or center-of-mass motion when useful.

21 Common mistakes

1. Assuming internal forces do zero work merely because they cancel in the net force.
2. Replacing actual external work with F_ext,net ⋅ ΔR_CM without justification.
3. Forgetting that pseudowork determines ΔK_CM, not generally ΔK.
4. Omitting relative kinetic energy.
5. Double counting a conservative internal force as both work and potential energy.
6. Summing pair potentials over both i,j and j,i instead of using each pair once.
7. Assuming zero net external force implies zero total kinetic energy.
8. Assuming a center of mass at rest means all particles are at rest.
9. Ignoring internal energy release in an explosion.
10. Applying one-particle energy formulas to a multi-particle system without defining the system boundary.

22 Practice exercises

1. Two particles have masses m₁ = 1 kg and m₂ = 2 kg with velocities 3 m∕s and −1 m∕s. Find V _CM and the total kinetic energy.
2. For the preceding system, calculate K_CM and K_rel and verify K = K_CM + K_rel.
3. Starting from v_i = V_CM + v_i′, derive the system kinetic-energy decomposition.
4. Show that ∑ _im_iv_i′ = 0.
5. For an internal pair satisfying F_ij = −F_ji, derive

   (119)

6. Explain why equal-and-opposite internal forces can do nonzero net internal work.
7. Two equal masses are connected by an ideal spring and move with no external force. Explain which energy terms can change and which center-of-mass quantity remains constant.
8. Derive K_rel = μv_rel² for two particles.
9. A two-fragment explosion starts from rest. If m₁ = 1 kg moves at 8 m∕s and m₂ = 4 kg, find the second fragment’s velocity and the final total kinetic energy.
10. In a uniform gravitational field, prove

    (120)

11. Explain the difference between actual external work and center-of-mass pseudowork.
12. A constant external force F moves one of two equal particles through distance d while the other remains fixed. Find the actual work, pseudowork, ΔK_CM, and ΔK_rel.
13. Give an example in which K_CM is constant while K_rel increases.
14. Give an example in which the system’s total momentum is zero but its total kinetic energy is not zero.
15. Explain how internal friction can reduce mechanical energy without violating total-energy conservation.

23 Summary

The total kinetic energy of a particle system is

(121)

The system work-energy theorem is

(122)

Internal forces can do net work even when they cancel from the net-force equation.

The kinetic energy decomposes as

(123)

For two particles,

(124)

The center-of-mass equation gives the pseudowork relation

(125)

This is not generally equal to the actual external work.

With conservative internal and external potentials,

(126)

These ideas provide the bridge from particle mechanics to collisions, rigid bodies, orbital two-body motion, and continuum systems.

The next article, M03-09, develops energy methods for rigid bodies.

References

References