Chapter 6 — Work Energy And Power

Definition: Work is the scalar quantity equal to the product of the component of a force along the direction of displacement and the magnitude of the displacement. It measures the energy transfer to or from a body by the action of a force.

SI unit: Joule (J) where 1 J = 1 N·m.

Work by a constant force: If a constant force F acts on a body and the body is displaced by a vector s, the work done is
W = F · s = F s cos θ,
where θ is the angle between the force and displacement vectors.

Work by a variable force: For a force that varies with position, work from r1 to r2 is the line integral
W = ∫_r1^r2 F(r) · dr.

Sign convention: Work can be positive, negative or zero.

• Positive: force has a component along displacement (e.g., gravity doing positive work when object falls).
• Negative: force has a component opposite to displacement (e.g., friction opposing motion).
• Zero: force is perpendicular to displacement (e.g., centripetal force in uniform circular motion; carrying a suitcase horizontally while lifting force is vertical).

Special cases and useful results:

• Work by gravity (near Earth): W_grav = - mg Δy = mg (y_i - y_f) where Δy = y_f - y_i.
• Work done to stretch/compress a spring (Hooke's law F = -kx): W = ∫₀^x kx' dx' = 1/2 k x² (work done by an external agent to stretch from 0 to x). The spring force does -1/2 k x² of work over that process.
• Work–energy theorem: Net work done on a particle equals the change in its kinetic energy: W_net = ΔK = (1/2) m v_f² - (1/2) m v_i².
• Conservative vs non-conservative forces: Work by a conservative force is path-independent and the work over any closed path is zero. Gravity and elastic spring forces are conservative; friction is non-conservative.

How to compute work (steps): 1) Resolve forces into components along displacement. 2) Use W = ∫ F · dr for variable forces or W = F s cos θ for constant forces. 3) Pay attention to sign conventions.

Definition: When a force varies with position, the work done in moving a particle from point A to B is the integral of the force along the path. For a small displacement dr, the infinitesimal work is dW = F · dr (scalar product).

One-dimensional case: If the force acts along the x-axis and depends on x, the work done moving from x1 to x2 is

W = ∫(x1 to x2) F(x) dx

Derivation idea (Riemann sum): Divide the path into small segments Δxi where the force is approximately constant F(xi). The total work ≈ Σ F(xi) Δxi. In the limit Δxi → 0 this sum becomes the integral above. Graphically this is the area under the F vs x curve between x1 and x2.

Vector form and sign: In vector form, W = ∫(path) F · dr. The sign of W depends on the angle between F and dr: positive when component of force is along displacement, negative when opposite. For motion in one dimension, a negative F(x) over the interval gives negative work (area below the x-axis).

Example: Work done by a spring: For Hooke's law, the spring (restoring) force is F(x) = -k x (x measured from natural length). Work done by the spring in moving from x1 to x2 is

W_spring = ∫(x1 to x2) (-k x) dx = -1/2 k (x2^2 - x1^2).

If we compress a spring from 0 to x, the external agent must do W_ext = +1/2 k x^2 (equal in magnitude and opposite in sign to the work done by the spring).

Work-energy theorem: The net work done by all forces on a particle equals the change in its kinetic energy: W_net = ΔK = 1/2 m v2^2 - 1/2 m v1^2. This remains valid when forces are variable; compute each force's work by integration and sum.

Conservative vs nonconservative: For conservative forces (e.g., gravity, spring), work between two points depends only on endpoints, not path. For nonconservative forces (e.g., friction), work generally depends on the path taken.

Units: SI unit of work is joule (J) = N·m.

Practical notes for solving problems:

• Identify the force as a function of position F(x) along the path.
• Choose correct limits x1 and x2 and sign convention.
• Compute integral W = ∫ F(x) dx or use known antiderivatives (polynomial, inverse-square, exponential, etc.).
• For vector motion in 2D/3D, compute W = ∫ F_x dx + ∫ F_y dy + ... or evaluate the line integral F · dr.

Common variable-force examples include springs, gravitational force varying with radial distance (inverse-square), nonuniform friction or drag that depends on position, and forces in elastic materials varying with extension.

Definition and Hooke's law: For an ideal (linear) spring, the restoring force exerted by the spring is proportional to its displacement from the natural (equilibrium) length: F_spring = -k x, where k is the spring constant (N/m), x is displacement measured from equilibrium, and the minus sign indicates the force is opposite to the displacement.

Work done by the spring force (derivation): Work done by the spring when its end moves quasi‑statically from position x1 to x2 is the integral of the spring force over displacement:

W_spring = ∫_{x1}^{x2} F_spring(x) dx = ∫_{x1}^{x2} (-k x) dx = -½ k (x2^2 - x1^2) = ½ k (x1^2 - x2^2).

This formula shows the work done by the spring depends only on the initial and final displacements (path independent) — a signature of a conservative force.

Elastic potential energy: We can define the elastic potential energy stored in a spring as U(x) = ½ k x^2 (choosing U(0)=0). The change in spring potential energy between x1 and x2 is ΔU = U(x2) - U(x1) = ½ k (x2^2 - x1^2). The relation between work by the spring and change in potential is W_spring = -ΔU.

Sign conventions and physical interpretation: If the spring goes from x1=0 to x2=+a (stretching), F_spring is negative while displacement is positive, so W_spring = -½k a^2: the spring does negative work (it absorbs energy). An external agent must do +½k a^2 work to stretch the spring slowly; this work is stored as elastic potential energy. Conversely, when the spring returns from +a to 0 it does positive work +½k a^2 on the object, releasing the stored energy.

Conservative nature and energy conservation: Because spring force is conservative, mechanical energy is conserved when only the spring and conservative forces act: ΔK + ΔU = 0. Thus work done by nonconservative forces equals change in (K + U).

Definition: Work done by gravity is the work performed by the gravitational force on a mass when it undergoes a displacement. Gravity is a conservative force, so the work done depends only on the initial and final vertical positions (heights), not on the path taken.

Vector form and basic formula: Let F_g = −mg ĵ (taking +y upward). For a displacement Δr = (y_B − y_A) ĵ,

W_g = ∫_A^B F·dr = (−mg) ∫_{y_A}^y_B dy = −mg(y_B − y_A) = mg(y_A − y_B).

Interpretation: If the object moves downward (y_B < y_A) the work done by gravity is positive (gravity does positive work and kinetic energy increases). If the object is lifted (y_B > y_A) gravity does negative work (it removes kinetic energy).

Relation with potential energy: Define gravitational potential energy near Earth as U = mgy (choice of zero arbitrary). Then

W_g = −ΔU = −(U_B − U_A) = m g (y_A − y_B).

Conservative property: For any closed path (returning to the same height) the net work done by gravity is zero: W_closed = 0. Work by gravity depends only on vertical displacement, not on the shape of the path.

Work for motion along an incline: If a block moves a distance s along an incline of angle θ and the vertical drop is h = s sinθ, then

W_g = mg h = mg s sinθ (sign positive if motion is downwards).

General (radial) gravitational field — extension: For large distances where g varies (Newtonian gravity), the work by gravity moving a mass from radial distance r_A to r_B is

W = ∫_{r_A}^r_B (−GMm/r^2) dr = GMm(1/r_B − 1/r_A).

Key points to remember:

• Work by gravity depends only on vertical height change: W = mg(y_A − y_B).
• Gravity does zero work for purely horizontal displacement at constant height.
• If an object falls freely from height y_A to y_B, the work done by gravity equals the loss in gravitational potential energy and equals the gain in kinetic energy (neglecting non‑conservative forces): W = ΔK = −ΔU.

Overview: Work by friction and other non-conservative forces is the mechanical work that cannot be recovered as potential energy because it is converted into internal energy (usually heat), sound or deformation. Unlike conservative forces (e.g., gravity, spring force), work done by non-conservative forces depends on the path taken.

General definition: For any force F acting along a path from point A to point B, the work is
W = ∫_A^B F · ds. Non-conservative forces give path-dependent values for this integral.

Work done by kinetic (sliding) friction: Kinetic friction acts opposite to the direction of instantaneous displacement. For a constant kinetic friction force f_k opposite to motion over a displacement s,

W_fric = − f_k s.

Using f_k = μ_k N (where N is the normal reaction), for a block on a horizontal surface N = mg and

W_fric = − μ_k m g s.

On an incline at angle θ where N = mg cos θ,

W_fric = − μ_k m g cos θ  s.

Integral form for variable or direction-dependent friction:

W_fric = −∫_path f_k(s) ds = −∫_A^B |f_fric(s)| ds,

which highlights path dependence: longer path → greater magnitude of negative work.

Non-conservative forces and mechanical energy: If non-conservative forces do work W_nc on a system, the change in mechanical energy (kinetic + potential) equals that work:

W_nc = ΔE_mech = ΔK + ΔU.

When friction is the only non-conservative force, W_fric < 0, so mechanical energy decreases and the magnitude |W_fric| is the energy converted to internal energy (heat):

Energy dissipated (as heat, etc.) = −W_fric = −W_nc.

Power dissipated by friction: Instantaneous power by friction P = F_fric · v. For kinetic friction opposite velocity, P = −f_k v (negative, meaning mechanical power removed).

Static friction note: Static friction does no work if the point of contact does not move relative to the surface (no slipping). Example: a wheel rolling without slipping has zero work by the static friction force on the center-of-mass translation (but internal deformations/rolling resistance can still dissipate energy).

Microscopic origin: Friction converts ordered mechanical energy into microscopic random motion of molecules (thermal energy) through surface interactions and deformation.

Key conceptual points:

• Work by friction is usually negative (it removes kinetic/mechanical energy).
• Non-conservative forces are path-dependent; conservative are path-independent and associated with potential energy.
• W_nc = ΔE_mech is a useful bookkeeping relation: if W_nc < 0 mechanical energy decreases; if W_nc > 0 mechanical energy increases (e.g., engine doing positive work).

Kinetic energy (KE) is the energy a body possesses due to its motion. For a particle of mass m moving with speed v, its kinetic energy K is defined as K = 1/2 m v². KE is a scalar and its SI unit is the joule (J).

Work done by a force: If a constant force F acts on a particle producing a displacement s in the direction of the force, the work done is W = F · s. For a variable force or a curved path, infinitesimal work dW = F · ds and the total work is the line integral W = ∫ F · ds.

Derivation of KE formula and the work–energy theorem (constant force, straight line):

Consider a particle of mass m acted on by a net constant force F causing acceleration a. By Newton's second law, F = m a. If the particle's speed changes from u to v while it moves through displacement s under the action of F, the work done by the net force is

W = F s = m a s.

Using the kinematic relation v² = u² + 2 a s, we get a s = (v² - u²)/2. Substituting,

W = m [(v² - u²)/2] = (1/2) m v² - (1/2) m u² = K_final - K_initial = ΔK.

This result is the work–energy theorem: the net work done on a particle by all forces equals the change in its kinetic energy. In differential/integral form for a general (possibly variable) net force F,

dW = F · ds = d(1/2 m v²) and W_net = ∫ F · ds = ΔK.

Sign and interpretation:

• Positive work (W > 0) increases kinetic energy (speed tends to increase).
• Negative work (W < 0) decreases kinetic energy (speed tends to decrease).
• If W = 0, kinetic energy is unchanged (e.g., force perpendicular to displacement).

Relation to conservation of mechanical energy: If the net work by non-conservative forces (like friction) is zero, the work done by conservative forces can be expressed as change in potential energy U with W_conservative = -ΔU. Then the work–energy theorem gives ΔK = -ΔU, or K + U = constant (mechanical energy conserved).

Applications: The theorem simplifies many problems because you can compute change in speed from work without solving the full force/acceleration differential equation. It is especially useful in braking, collisions (work-energy used with impulse-momentum), ramps, springs (elastic potential), and energy-transfer calculations.

Note: For extended bodies or rotating systems, there are analogous forms: translational KE = 1/2 M V_cm², rotational KE = 1/2 I ω², and work–energy relations include both translational and rotational kinetic energies.

Definition: Potential energy (PE) is the energy possessed by a body due to its position or configuration in a force field (usually a conservative force field). It is a scalar quantity measured in joules (J).

Physical idea: When a conservative force (for example gravity or an ideal spring force) can do work as an object changes position, we can associate a potential energy function U such that the work done by that force when moving the object between two points equals the decrease in U. In symbols, for a conservative force F, the change in potential energy when the system moves from A to B is ΔU = U(B) − U(A) = −W_cons(A→B).

Reference level: Only differences in potential energy have physical meaning. You choose a zero of potential energy (reference level) conveniently (e.g., ground, lowest point). Changing the reference shifts U by a constant but does not affect dynamics.

Relation to force: In one dimension, the conservative force is the negative spatial derivative of potential energy: F(x) = −dU/dx. In three dimensions, F = −∇U (the negative gradient). This relation tells how the shape of U(x) controls motion: the force pushes objects toward lower U.

Energy conservation: In the absence of non-conservative forces, mechanical energy is conserved: E = K + U = constant, where K is kinetic energy. As U increases, K decreases and vice versa.

Equilibrium and stability: Points where dU/dx = 0 are equilibrium points. If U has a local minimum there, the equilibrium is stable (small displacements produce restoring forces); if U has a local maximum, the equilibrium is unstable. Near a stable minimum U(x) ≈ U0 + (1/2)k(x−x0)^2, giving simple harmonic motion with effective spring constant k = U''(x0).

Definition: A force is conservative if the work it does on a particle moving between two points is independent of the path taken. Equivalently, the work done by a conservative force around any closed path is zero. A force is non-conservative if the work depends on the path and the work done around a closed path is not zero.

Key properties of conservative forces

• Path independence: W(A→B) is the same for all paths joining A and B.
• Zero work over a closed loop: ∫_closed F·dr = 0.
• Existence of potential energy V(r): F = −∇V (so W(A→B) = V(A) − V(B)).
• Mathematical condition (in a simply connected region): curl F = 0 (∇ × F = 0).

Key properties of non-conservative forces

• Path dependent: W(A→B) depends on the path taken between A and B.
• Nonzero work over closed loop: ∫_closed F·dr ≠ 0 (it often removes mechanical energy).
• No single-valued potential energy function exists for the force.
• Typically dissipative: converts mechanical energy into thermal/internal energy (friction, air drag).

Energy viewpoint

• For conservative forces, mechanical energy (kinetic + potential) is conserved: E_mech = K + V = constant (if only conservative forces act).
• If non-conservative forces do work W_nc, the change in mechanical energy equals that work: ΔE_mech = W_nc (usually negative for dissipative forces).

Simple examples (conceptual): Gravity, electrostatic force and ideal spring force are conservative. Friction, air resistance and viscous damping are non-conservative.

Definition: The law of conservation of mechanical energy states that for a system on which only conservative forces (like gravity or ideal spring forces) act, the total mechanical energy — the sum of kinetic energy (K) and potential energy (U) — remains constant.

Why it holds: From the work–energy theorem, the net work done on a particle equals the change in its kinetic energy: W_net = ΔK. If the only forces are conservative, the work done by those forces can be expressed as minus the change in potential energy: W_cons = -ΔU. Hence ΔK = -ΔU, or Δ(K + U) = 0, so K + U = constant.

Derivation (single particle, conservative forces only):

• Work–energy theorem: W_net = ΔK.
• For conservative forces: W_cons = -ΔU.
• Thus ΔK = -ΔU → Δ(K + U) = 0 → K + U = constant.

Interpretation: Mechanical energy is continuously exchanged between kinetic and potential forms. At an instant where potential energy is maximum, kinetic energy is minimum (and vice versa), but their sum stays fixed. The reference zero for potential energy is arbitrary — only differences in U matter.

When it fails or is modified: If non-conservative forces (friction, air resistance, inelastic deformation) do work, mechanical energy is not conserved; some mechanical energy is transformed into thermal energy or other forms. General relation: ΔE_mech = W_nc, or E_mech_final = E_mech_initial + W_nc (W_nc is typically negative for dissipative forces).

Common potential energies: Near Earth's surface: U_g = mgh. For an ideal spring: U_s = (1/2) k x^2. Kinetic energy: K = (1/2) m v^2.

Definition: Power is the rate at which work is done or energy is transferred. It quantifies how quickly energy is converted from one form to another.

Average and Instantaneous Power:

• Average power over a time interval Δt: P_avg = W / Δt, where W is work done in time Δt.
• Instantaneous power is the limit as Δt → 0: P = dW/dt.

Mechanical expression (translational): If a force F acts on a particle moving with instantaneous velocity v, the instantaneous power delivered by the force is P = F · v = F v cosθ, where θ is the angle between F and v. This follows because dW = F · ds and v = ds/dt, so P = dW/dt = F · v.

Relation to kinetic energy: For a particle of mass m, the rate of change of kinetic energy equals the net power input: d/dt(1/2 m v^2) = m v a = F_net · v = P_net. This ties power to how fast the kinetic energy changes.

Rotational analogue: For rotation, torque τ and angular speed ω give power: P = τ ω. If a set of forces produces a torque τ and the body rotates with angular speed ω, the power transferred is τ times ω.

Sign convention: If force has a component opposite to velocity, P is negative (work done by the object on the agent). Positive P means the force does work on the object.

Units: SI unit of power is the watt (W): 1 W = 1 J/s. Common unit: horsepower (hp), 1 hp ≈ 746 W.

Useful notes:

• For constant force along the displacement direction and constant speed v: P = F v.
• Electrical expressions often used: P = V I = I^2 R = V^2 / R.
• Mechanical power rating (e.g., motors) gives the maximum useful energy transfer per unit time; efficiency = (useful power out)/(power in).

Typical classroom derivations/problems: Show that a person lifting a mass m at constant speed v requires power P = mg v. Show that for a car with constant engine power P, the acceleration decreases as v increases because d/dt(1/2 m v^2) = P gives v dv/dt = P/m.

Basic idea: Work measures the energy transfer when a force acts through a displacement. In two or three dimensions both force and displacement are vectors, and only the component of force along the displacement does work. Work is a scalar quantity; SI unit: joule (J).

Vector (dot) product definition: If a constant force F acts while an object undergoes a displacement Δr, the work done is the dot product

W = F · Δr = |F| |Δr| cos θ

where θ is the angle between the force vector and the displacement vector. In component form (3D):

W = F_xΔx + F_yΔy + F_zΔz

Work by a variable force (general path): If the force varies in space or the object moves along a curved path, work is given by the line integral of the force along the path C:

W = ∫_C F · d r

Interpretation: break the path into infinitesimal displacements d r = (dx, dy, dz); the infinitesimal work is dW = F · d r and total work is the integral of dW along the path.

Conservative vs non-conservative forces: For conservative forces (e.g., gravity, ideal spring), work depends only on the initial and final positions, not on the path. For gravity near Earth (taking +y upward): W_gravity = -m g Δy (work done by the gravitational force on the object). For non-conservative forces (e.g., kinetic friction), work depends on the path and usually converts mechanical energy into heat.

Key practical points:

• When the force is perpendicular to displacement, W = 0 (e.g., centripetal force acting perpendicular to instantaneous displacement).
• If you carry an object horizontally at constant height, the vertical force you apply does no net work on the object (displacement vertical = 0); only horizontal components matter.
• Work can be split: if multiple forces act, total work = sum of works by each force (work is additive).

Short worked idea (2D): A force F of magnitude 50 N acts at 30° above the horizontal and displaces an object by 4 m horizontally. Take displacement vector = (4,0). The work is W = F cos30° × 4 = 50 × (√3/2) × 4 ≈ 173.2 J.

Why use components: In 2D/3D problems it is often easiest to resolve F into components and use W = F_xΔx + F_yΔy (+ F_zΔz in 3D). For curved motion, parametrize the path (x(t),y(t),z(t)) and compute W = ∫ (F_x dx + F_y dy + F_z dz).

Summary: Work in 2D and 3D is computed using the scalar (dot) product of force and displacement. For variable forces or curved paths use the line integral. Distinguish conservative forces (path-independent) from non-conservative ones (path-dependent).

Overview: Graphical interpretation links physical quantities to geometric features (area under curve, slope) so you can read work, energy and power directly from plots. In the Work–Energy chapter the most useful graphs are Force vs displacement (F–x), Potential energy vs displacement (U–x), Power vs time (P–t) and Kinetic/Energy vs time graphs. Understanding which geometric property (area, slope, intercept) corresponds to which physical quantity is the key.

What to read from common graphs:

• F vs x (force vs displacement): Area under the curve between x1 and x2 gives work done by that force: W = ∫(x1 to x2) F(x) dx. For a constant force F, W = FΔx. For variable force, calculate area (geometric shapes or integral).
• U vs x (potential energy vs displacement): Slope gives force with sign: F(x) = -dU/dx. Minima of U correspond to stable equilibrium; difference U(x2) − U(x1) is change in potential energy.
• P vs t (power vs time): Area under the P–t curve from t1 to t2 equals energy delivered or consumed: ΔE = ∫(t1 to t2) P(t) dt.
• K vs x or K vs t: Changes in kinetic energy shown on the graph connect to work by the work–energy theorem: ΔK = W_net.

Sign conventions and interpretation: Positive area means force has a component along displacement direction (work done by the force on the object); negative area means the force opposes motion (work done by the object on the force). For U–x, if dU/dx > 0 then the force is negative (opposes increasing x).

Problem‑solving techniques (stepwise):

1. Identify which graph you have and what quantity you need (work, change in potential, energy delivered, force at a point).
2. Decide whether the required operation is area under curve (integral) or slope of curve (derivative).
3. If area is needed, split the region into simple geometric shapes (rectangles, triangles, trapezoids) or set up an integral when the curve is given analytically.
4. Keep track of signs: work by a force along +x is positive; if displacement is opposite the force, area is negative.
5. Use the work–energy theorem (W_net = ΔK) or energy conservation (K + U = constant) as shortcuts when multiple forces are present.
6. Check units (N·m = J for work/energy; W·s = J for power-time areas; N = J/m for slope of U–x if inverted sign).
7. For numerical graphs, approximate a curve by small trapezoids (trapezoidal rule) if no analytic form is given.

Common pitfalls: confusing area under F–t (impulse) with F–x (work); forgetting negative sign when using U–x; mixing up which axis corresponds to integration variable.

What is a potential energy curve? A potential energy curve U(x) is a graph of the potential energy of a system as a function of a generalized coordinate x (position). For systems with conservative forces, the force is determined by the slope of this curve: F(x) = -dU/dx. The system's motion and equilibrium positions are understood from the shape of U(x).

Equilibrium points

• An equilibrium point x0 satisfies dU/dx|_{x=x0} = 0 (horizontal tangent on the U vs x curve).
• Stability criteria (using second derivative):
  • Stable equilibrium: d^2U/dx^2|_{x0} > 0. U(x) has a local minimum; small displacements produce a restoring force back toward x0.
  • Unstable equilibrium: d^2U/dx^2|_{x0} < 0. U(x) has a local maximum; small displacements grow away from x0.
  • Neutral (marginal) equilibrium: d^2U/dx^2|_{x0} = 0 (higher derivatives decide behavior); U is flat near x0.

Small oscillations about a stable equilibrium

Near a stable minimum x0 expand U(x) in a Taylor series: U(x) ≈ U(x0) + 1/2 U''(x0) (x - x0)^2. The system behaves like a simple harmonic oscillator with effective spring constant k = U''(x0). For a mass m, angular frequency ω = sqrt(k/m) and period T = 2π sqrt(m/k).

Energy viewpoint and turning points

For a conservative system with total energy E = K + U(x), motion is allowed only where U(x) ≤ E. Points where U(x) = E are turning points (kinetic energy zero). Plotting a horizontal line E on the U(x) graph visually shows accessible regions.

Physical interpretation of force

The force at each x is opposite the slope of U(x): if U increases with x (dU/dx > 0) the force is negative (pulls left); if U decreases with x (dU/dx < 0) the force is positive (pushes right).

What is a system of particles? A system of particles is a collection of many particles (or bodies) considered together. Each particle i has mass m_i, position r_i(t) and velocity v_i(t). We study the motion of the whole system through quantities like total mass, total momentum and centre of mass (CM).

Centre of mass (CM): The position vector of CM is

R = (1/M) Σ m_i r_i,    where M = Σ m_i

Velocity and acceleration of CM:

V = dR/dt = (1/M) Σ m_i v_i

a_CM = dV/dt

The total linear momentum of the system is P = M V = Σ m_i v_i.

Internal and external forces: Each particle experiences forces. Forces due to agents outside the system are external forces (F_ext). Forces between particles of the system are internal forces (F_ij is force on i due to j). By Newton's 3rd law for pairwise interactions, F_ij = −F_ji (for equal and opposite forces along the line joining particles).

Equation of motion for the CM: Summing Newton's second law for all particles gives

M a_CM = Σ F_ext + Σ Σ F_ij.

Because Σ_i Σ_j F_ij = 0 (internal forces cancel in pairs), we get the important result

M a_CM = Σ F_ext.

Thus the net external force determines the acceleration of the CM; internal forces cannot change the motion of the CM. If Σ F_ext = 0, the total momentum P is conserved.

Kinetic energy: decomposition: The total kinetic energy of the system can be split into kinetic energy of the CM motion and kinetic energy relative to the CM:

K_total = Σ (1/2 m_i v_i^2) = (1/2) M V^2 + Σ (1/2 m_i u_i^2)

where u_i = v_i − V is the velocity of particle i relative to the CM. The first term is the kinetic energy of the whole system moving as if all mass were at the CM; the second is internal (or relative) kinetic energy.

Work–energy for a system: Multiplying each particle's equation of motion by its velocity and summing yields the rate form of work–energy theorem. In integrated form:

ΔK_total = W_ext + W_int.

Internal forces can do work that changes relative kinetic energy or can be associated with changes in internal potential energy. If internal forces are conservative (so we can define internal potential energy U_int), then

Δ(K_total + U_int) = W_ext.

So when only conservative internal forces act and external work is zero, mechanical energy (K_total + U_int) is conserved.

Consequences and typical behaviors:

• Momentum conservation: If resultant external force = 0, total momentum P is constant.
• Internal forces cannot change motion of CM, but they can redistribute energy among particles and convert kinetic energy to internal energy (e.g., heating during inelastic collisions).
• For constant mass systems, motion of CM is governed solely by external forces; for variable-mass systems (rockets, raindrops collecting mass) additional terms appear—treat separately.

Summary: The key ideas are (1) define CM and total momentum, (2) internal forces cancel in the net force so CM motion is controlled only by external forces, (3) total kinetic energy splits into CM motion and internal motion, and (4) internal forces may change internal/relative energy but not the CM momentum; for conservative internal forces total mechanical energy plus potential is conserved when no external work is done.

Overview
This topic covers the SI units and dimensional formulas of physical quantities related to work, energy and power, how to use dimensional analysis to check equations, and the important formulas used in these concepts.

SI units and fundamental idea
The SI base units relevant here are: mass (kg), length (m) and time (s). Derived units used often in this chapter are:
1 J (joule) = 1 N·m = 1 kg·m²·s⁻² (unit of work and energy)
1 W (watt) = 1 J·s⁻¹ = 1 kg·m²·s⁻³ (unit of power)

Dimensions and dimensional formula
A dimensional formula expresses a quantity in terms of base dimensions M (mass), L (length), T (time). Dimensional homogeneity: both sides of any physically correct equation must have the same dimensions.

Common dimensional formulas (useful in this chapter)

• Displacement, x: [L]
• Velocity, v: [L T⁻¹]
• Acceleration, a: [L T⁻²]
• Force, F: [M L T⁻²]
• Work / Energy, W or E: [M L² T⁻²]
• Power, P: [M L² T⁻³]
• Momentum, p: [M L T⁻¹]
• Spring constant, k: [M T⁻²]

Dimensional analysis – uses and limitations
Uses: check correctness of derived equations, find form of a relation up to a dimensionless constant, identify possible dependent variables. Limitations: cannot determine dimensionless constants (like 1/2), cannot distinguish quantities with same dimensions (e.g., energy and torque), and cannot give numerical coefficients or functional forms with transcendental functions.

Important conceptual points

• Work is scalar: W = F·s·cosθ (dot product of force and displacement).
• Positive work adds energy to a body; negative work removes energy.
• Work done by a variable force is the area under the force–displacement curve: W = ∫ F(x) dx.
• Work–energy theorem: net work done on a particle = change in its kinetic energy, W_net = ΔK.
• Mechanical energy (K + U) is conserved if only conservative forces act (no non‑conservative work like friction).
• Power measures rate of doing work: instantaneous P = dW/dt = F·v.