7.3 Apply work–energy to a system of particles | University of Alberta EN PH 131: Engineering Mechanics: Dynamics

University of Alberta EN PH 131: Engineering Mechanics: Dynamics

Systems of Particles

Adding particle energy changes and accounting for work consistently

Work–energy relates the work done by forces to changes in motion. For a system of particles, calculate each particle’s kinetic-energy change and add the results. The system’s total work includes work done by external forces and, when relevant, internal forces between particles. Alternatively, work by conservative forces such as gravity or an ideal spring can be represented by a change in potential energy. Keep these two accounting methods distinct to avoid double-counting.

What you will learn

Define a particle system, observer, reference frame, and initial and final states.
Calculate a system’s total kinetic energy by adding the energies of its particles.
Apply work–energy while distinguishing total force work from an equation using potential energy.
Account for gravity and an ideal spring without counting their effects twice.
Check work–energy results using signs, SI units, and physical limits.

1. Define the system, frame, and energy

A system is the collection of particles selected for analysis. A particle is represented by its mass and motion, without considering its size. The observer measures motion in a reference frame. Unless a problem says otherwise, use a frame fixed to the ground. Define coordinates and positive directions—for example, horizontal

x

positive right and vertical

y

positive upward—before assigning signs to displacements or velocity components.

For particle

i

, kinetic energy is one-half its mass times its speed squared. The total kinetic energy is the sum over all particles. Speed is the magnitude of velocity, so a particle moving left or downward still has nonnegative kinetic energy. Measure all speeds in the same stated frame.

The work–energy principle follows by applying the particle work–energy relation to every particle and adding the equations. For the selected system, the change in total kinetic energy equals the total work done on its particles by all forces. This includes external forces exerted from outside the system and internal forces exerted between particles in it. Internal forces must not be discarded automatically: their work can transfer energy between particles or store energy in an interaction such as an ideal spring.

T=\sum_i \frac12 m_i v_i^2

State which particles belong to the system and which frame measures their speeds.
Add the kinetic energies of individual particles; do not assign one speed to the entire system unless justified.
The total-work form includes work by all forces on the particles.

2. Work and the two energy-accounting forms

Work is the line integral of force dotted with the displacement along the path. For a constant force, it is the force component along the displacement multiplied by the displacement distance. A force perpendicular to the displacement does no work. When force changes along a path, use the integral rather than assuming the force stays constant.

The direct work–energy statement says that the change in total kinetic energy equals the total work of all forces on all particles. This is the most general form used here. For example, if an internal spring force acts on two particles, its work on both particles belongs in the total work.

If a force is conservative, its work can instead be represented by a potential-energy change: its work equals the negative change in that potential energy. For gravity near Earth’s surface, gravitational potential energy is mass times gravitational acceleration times height. For an ideal spring, spring potential energy is one-half stiffness times extension squared. When using this form, leave the conservative force’s work out of the explicit work term. Total force work and potential-energy change are alternative accounts for the same conservative force, not two contributions to add together.

T_1+V_1+W_{\mathrm{nc}}=T_2+V_2

Use the total-work equation when accounting directly for force work.
When replacing conservative-force work with potential energy, use the negative potential-energy change.
Include work by non-conservative forces explicitly in the potential-energy form.

3. A reliable solution sequence

First define the system, observer, reference frame, axes, and positive directions. Identify the initial and final states and list the given masses, speeds, positions, forces, and any spring data. Sketch the particles and relevant displacement, force, or energy changes when a picture helps clarify the signs.

Next choose an accounting method. In the direct form, sum the work done by all forces on all particles. In the potential-energy form, include changes in the potential energies of conservative interactions and the work of any non-conservative forces. Write the equation symbolically before substituting values where practical. Work–energy is well suited to finding a speed or energy change between two states, but by itself it does not determine elapsed time.

Finally check the result. Work and energy have units of joules, with one joule equal to one newton-metre. In a constant-force calculation, force times distance must have those units. Check whether each force’s work sign matches its direction relative to displacement. A speed must be nonnegative, and the result should match the stated initial and final conditions. A useful limiting check is that a force perpendicular to the displacement does zero work.

W_{1\to2}=\int_1^2 \mathbf{F}\cdot d\mathbf{r}

Define the system and states before writing an energy balance.
Keep force-work signs consistent with each particle’s displacement.
Check dimensions, directions, and whether the chosen accounting method counts each energy transfer once.

4. Interpreting conservation for particles

When only conservative forces do work and their potential energies are included, the sum of kinetic and potential energy at the initial state equals that at the final state. This does not mean each particle keeps the same kinetic energy. One particle can gain kinetic energy while another loses it, and potential energy can change as the system configuration changes.

For a system that includes an ideal spring, its potential energy is part of the system’s energy account. For a system moving near Earth, gravitational potential energy can be included when height changes. If an external force or friction does work, that work must also be included in the balance. Always state which forces are represented by potential energy and which remain explicit work terms.

T_1+V_1=T_2+V_2

Conservation applies to the complete energy balance, not to each particle’s kinetic energy separately.
Include spring or gravitational potential-energy changes only when those interactions are part of the chosen account.
Include non-conservative work explicitly.

Worked example

Net work on two particles

Two particles move on a horizontal, frictionless track in a ground-fixed frame. Particle A has mass

2.0\,\mathrm{kg}

and its speed changes from

1.0\,\mathrm{m/s}

to

3.0\,\mathrm{m/s}

. Particle B has mass

1.0\,\mathrm{kg}

and its speed changes from

4.0\,\mathrm{m/s}

to

2.0\,\mathrm{m/s}

. What is the total work done on the two-particle system?

Define the system and states
Take A and B as the system, measured in the ground-fixed frame. Choose $+x$ to the right. The initial state is before the speed changes; the final state is after them. Apply total work to the sum of the two particles’ kinetic-energy changes.
Calculate the total change
Kinetic energy depends on speed squared, so B’s direction does not make its energy negative. A gains $8.0\,\mathrm{J}$ and B loses $6.0\,\mathrm{J}$ .
$\Delta T=\frac12(2.0)(3.0^2-1.0^2)+\frac12(1.0)(2.0^2-4.0^2)=2.0\,\mathrm{J}$
Relate energy change to work
By work–energy, the total work of all forces on the particles equals their total kinetic-energy change. The net work is positive because the system’s total kinetic energy increased.
$W_{1\to2}=\Delta T=2.0\,\mathrm{J}$

Answer: The total work done on the two-particle system is

2.0\,\mathrm{J}

.

Check: The initial total kinetic energy is

9.0\,\mathrm{J}

and the final total is

11.0\,\mathrm{J}

. Their difference is

2.0\,\mathrm{J}

. Each term has units

\mathrm{kg\,m^2/s^2}=\mathrm{J}

.

Worked example

Two particles descending under gravity

A

1.5\,\mathrm{kg}

particle and a

0.50\,\mathrm{kg}

particle each descend vertically by

2.0\,\mathrm{m}

near Earth’s surface. Their initial speeds are

1.0\,\mathrm{m/s}

and

2.0\,\mathrm{m/s}

, respectively. Neglect air resistance and assume both continue downward. Find each final speed.

Gravity during descent

Weight and displacement point downward, so gravity does positive work.

Set the frame and states
Use a ground-fixed frame with $+y$ upward. The system is the two particles; the initial state is before descent and the final state is after each has moved downward by $2.0\,\mathrm{m}$ . Thus the vertical displacement is negative, while the weight is also directed downward.
$\Delta y=-2.0\,\mathrm{m}$
Apply work–energy to each particle
Use the direct work form for gravity. Weight and displacement have the same direction, so gravity’s work is positive. For each particle, its kinetic-energy increase equals the work of its own weight.
$v_f^2=v_1^2+2g(2.0\,\mathrm{m})$
Calculate the final speeds
Use $g=9.81\,\mathrm{m/s^2}$ . Take the positive square root because the requested quantities are speeds; the stated motion direction is downward.
$v_{Af}=\sqrt{1.0^2+2(9.81)(2.0)}=6.34\,\mathrm{m/s},\qquad v_{Bf}=\sqrt{2.0^2+2(9.81)(2.0)}=6.58\,\mathrm{m/s}$

Answer: The final speeds are approximately

6.34\,\mathrm{m/s}

for the

1.5\,\mathrm{kg}

particle and

6.58\,\mathrm{m/s}

for the

0.50\,\mathrm{kg}

particle, both downward.

Check: The system’s total kinetic-energy increase is

39.24\,\mathrm{J}

, equal to gravity’s work

(1.5+0.50)(9.81)(2.0)

. The quantity under each square root has units of

\mathrm{m^2/s^2}

, as required.

Worked example

Spring energy transferred to two carts

Two carts of masses

1.0\,\mathrm{kg}

and

3.0\,\mathrm{kg}

rest on a level, frictionless track and are connected by an ideal spring with stiffness

200\,\mathrm{N/m}

. The spring is stretched

0.10\,\mathrm{m}

from its unstretched length and the carts are released from rest. Find their total kinetic energy when the spring reaches its unstretched length.

Carts and spring

The spring’s stored energy becomes total cart kinetic energy.

Define the system and states
Take both carts and the ideal spring as the system in a ground-fixed frame, with $+x$ to the right. Initially both carts are at rest and the spring is stretched; finally the spring is unstretched. The level track’s gravity and normal forces do no work along the motion.
Use potential energy
Use the potential-energy form rather than adding spring work separately. The initial spring potential energy becomes the carts’ total kinetic energy. There is no initial kinetic energy and no spring potential energy at the final state; no non-conservative force does work along the track.
$T_1+V_{s1}=T_2+V_{s2}$
Substitute the spring data
The initial extension is $0.10\,\mathrm{m}$ and the stiffness is $200\,\mathrm{N/m}$ . This gives the sum of the two cart kinetic energies, even though their individual speeds are not requested.
$T_2=\frac12(200)(0.10)^2=1.0\,\mathrm{J}$

Answer: The carts’ total kinetic energy is

1.0\,\mathrm{J}

when the spring reaches its unstretched length.

Check: The spring-energy units are

(\mathrm{N/m})(\mathrm{m^2})=\mathrm{N\,m}=\mathrm{J}

. If the initial extension were zero, the stored spring energy and resulting total kinetic energy would both be zero.

Common mistakes and how to avoid them

Assuming internal forces always do zero total work.

Correction: Internal forces can transfer energy between particles or store energy in a spring. Include their work in the direct form or include the corresponding potential energy.

Using signed velocity as though kinetic energy could be negative.

Correction: Kinetic energy uses speed squared and is nonnegative regardless of direction.

Including a conservative force’s work and also including its potential-energy change.

Correction: Use either that force’s work or the negative change in its potential energy, not both.

Using total mass with one particle’s speed when the particles have different speeds.

Correction: Calculate each particle’s kinetic energy with its own mass and speed, then add the results.

Lesson summary

A system’s total kinetic energy is the sum of the kinetic energies of its particles.
The change in total kinetic energy equals total force work when work by all forces on the particles is included.
Internal work does not automatically cancel.
A conservative force’s work may be replaced by the negative change in potential energy; do not count both.
Check system boundaries, signs, SI units, states, and whether the result matches the stated motion.

Check your understanding

Question 1

Two particles have kinetic-energy changes of

+5\,\mathrm{J}

and

-2\,\mathrm{J}

. What is the total work on the system?

$3\,\mathrm{J}$
$7\,\mathrm{J}$
$-3\,\mathrm{J}$
$-7\,\mathrm{J}$

Show answer and explanation

3\,\mathrm{J}

The total change is

5\,\mathrm{J}+(-2\,\mathrm{J})=3\,\mathrm{J}

, so total force work is

3\,\mathrm{J}

.

Question 2

A particle moves horizontally while its weight acts vertically. What work does its weight do?

Zero
Positive work
Negative work
Work equal to the particle’s kinetic energy

Show answer and explanation

Zero

Weight is perpendicular to the horizontal displacement, so its force–displacement dot product is zero.

Question 3

An ideal spring initially stores

4\,\mathrm{J}

and is the only energy source for carts initially at rest on a frictionless level track. What is their total kinetic energy when the spring is unstretched?

$4\,\mathrm{J}$
$0\,\mathrm{J}$
$8\,\mathrm{J}$
It cannot be determined without the cart masses

Show answer and explanation

4\,\mathrm{J}

The spring potential energy decreases by

4\,\mathrm{J}

and becomes total cart kinetic energy. The masses are needed for individual speeds, not for this total energy.

Key terms

System: The chosen collection of particles whose energy changes are analyzed.
Work: Energy transferred by a force as its point of application moves; its sign depends on the force component along the displacement.
Internal force: A force exerted by one particle in the selected system on another particle in that same system.
Potential energy: Energy assigned to a conservative interaction, such as gravity or an ideal spring, based on the system’s configuration.

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7.3 · Apply work–energy to a system of particles

University of Alberta EN PH 131: Engineering Mechanics: Dynamics

Adding particle energy changes and accounting for work consistently

What you will learn

1. Define the system, frame, and energy

2. Work and the two energy-accounting forms

3. A reliable solution sequence

4. Interpreting conservation for particles

Net work on two particles

Two particles descending under gravity

Spring energy transferred to two carts

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through EN PH 131

About this lesson and its review