Kinetics of a Particle:
Work and Energy
This chapter recasts Newton's second law into a scalar statement linking force, velocity, and displacement. By integrating the tangential equation of motion, we obtain the principle of work and energy — a tool that sidesteps acceleration entirely when those three quantities are what a problem actually gives you.
Learning objectives
- Explain work as a force–displacement integral and interpret its sign physically.
- Select work–energy rather than force–acceleration or impulse–momentum when the known information is displacement-based.
- Calculate speeds and stopping distances with applied, gravitational, elastic, and friction work.
- Study problems that involve power and efficiency.
- Distinguish conservative potential-energy changes from nonconservative work.
- Verify units, signs, friction assumptions, contact, and endpoint admissibility.
14.1The Work of a Force
A force does work on a particle only when the particle undergoes a displacement in the direction of the force. If a force F moves a particle through a differential displacement \(d\mathbf{r}\) of magnitude \(ds\), and \(\theta\) is the angle between the force and the displacement, the differential work is a scalar.
Work of a Force
The work done by a force is the product of the force magnitude and the component of displacement along the force — or equivalently, the displacement times the force component along the displacement. Work is positive when \(0^\circ \le \theta < 90^\circ\), negative when \(90^\circ < \theta \le 180^\circ\), and zero when the force is perpendicular to the displacement or acts at a fixed point.
The Joule
In SI units, work is measured in joules. One joule is the work done by a one-newton force moving through one metre in the force's direction: \(1\ \text{J} = 1\ \text{N}\cdot \text{m}\).
Work of a Variable Force
When F and \(\theta\) vary with position, the total work over a finite path is found by integration. Graphically, it is the area under a plot of \(F\cos\theta\) versus \(s\).
Work of a Constant Force Along a Straight Line
Work of a Weight
The work of a constant weight depends only on the vertical displacement, not on the path taken. It is negative for upward motion and positive for downward motion.
Work of a Spring Force
For a linear elastic spring (\(F_s = ks\)), the work done on the attached particle as the stretch changes from \(s_1\) to \(s_2\) equals the negative change in the trapezoidal area under the \(F_s = ks\) line.
Sign Convention for Spring Work
Avoid sign errors by comparing the direction of the spring force on the particle with the particle's displacement: same sense → positive work; opposite sense → negative work.
Students lose the most marks here on spring work. Always sketch the spring's initial and final stretch separately before plugging into Eq. 14–4 — and remember \(s\) is measured from the unstretched position, never from the floor or wall.
14.2Principle of Work and Energy
Integrating the tangential equation of motion \(F_t = ma_t\) using the kinematic relation \(a_t = v\,dv/ds\) yields a scalar statement: the work done by all forces equals the change in the particle's kinetic energy.
Kinetic Energy
Kinetic energy, \(T = \tfrac{1}{2}mv^2\), is a scalar with units of joules. Unlike work, it is always positive regardless of the direction of motion.
For a particle moving from position 1 to position 2 under any system of forces, the particle's initial kinetic energy plus the work done by all forces equals its final kinetic energy.
\[ T_1 + \textstyle\sum U_{1\text{-}2} = T_2 \]Equivalently \(\sum U_{1\text{-}2} = \tfrac{1}{2}mv_2^2 - \tfrac{1}{2}mv_1^2\) (Eqs. 14–6, 14–7). This is an integrated form of \(F_t = ma_t\), convenient whenever a problem involves force, velocity, and displacement.
Procedure for Analysis
- Draw a free-body diagram to identify all forces that do work along the path.
- Apply \(T_1 + \sum U_{1\text{-}2} = T_2\); kinetic energy is always positive.
- Variable forces must be integrated (area under the force–displacement curve).
- Weight: \(U_W = \pm W\,\Delta y\) (positive moving downward). Spring: \(U_s\) from Eq. 14–4.
If a problem asks for speed or distance and gives you a force, reach for work–energy before the equation of motion. It collapses a two-step kinematics problem into one equation — because it often removes time and acceleration from the calculation.
14.3Work and Energy for a System of Particles
Because work and energy are scalars, the principle extends to a system of particles by algebraic summation over external and internal forces.
Internal Forces in Rigid Bodies
For a translating rigid body (or connected translating bodies), internal forces occur in equal, opposite, collinear pairs and cancel. In non-rigid bodies, particles move along different paths and some energy is dissipated as heat or stored through deformation.
Work of Friction Caused by Sliding
When a block slides over a rough surface, the resultant friction force is \(\mu_k N\), but localized deformations mean the friction force's actual displacement \(s'\) is less than the block's displacement \(s\). The term \(\mu_k N\,s\) therefore represents both the external work and internal work converted to heat.
Friction Is Non-Conservative
The external work of friction is \(\mu_k N\,s'\), while \(\mu_k N(s-s')\) becomes internal energy that raises the block's temperature. In this text, problems are handled where such losses need not be separately accounted for.
14.4Power and Efficiency
Power
Power is the time rate of doing work. Two machines may perform the same work, but the one with greater power finishes sooner. Power is a scalar; its SI unit is the watt, \(1\ \text{W} = 1\ \text{J/s} = 1\ \text{N}\cdot \text{m/s}\).
Mechanical Efficiency
Efficiency is the ratio of useful power output to power input. Because friction always consumes some energy within a machine, efficiency is always less than 1.
Procedure for Analysis
- Find the external force F producing the motion (use \(\sum F = ma\) if accelerating).
- Compute \(P = \mathbf{F}\cdot \mathbf{v} = Fv\cos\theta\) using the velocity at the point of application.
- Alternatively, \(P_{\text{avg}} = \Delta U/\Delta t\).
14.5Conservative Forces & Potential Energy
Conservative Force
A force is conservative if the work it does is independent of the path and depends only on the initial and final positions. Weight and spring forces are conservative; friction is not, because its work grows with path length and is dissipated as heat.
Energy & Potential Energy
Energy is the capacity to do work. Kinetic energy is associated with motion; potential energy is associated with position, measured from a fixed datum. It quantifies the work a conservative force will do when the particle returns to the datum.
Potential Function
When both gravity and springs act, the total potential energy is the algebraic sum \(V = V_g + V_e\). The work of a conservative force equals the drop in this function.
Criterion for a Conservative Force
Equation 14–19 relates a force to its potential function and provides a mathematical test: a force is conservative if it can be written as the negative gradient of a scalar potential. For weight, \(F_y = -\partial(Wy)/\partial y = -W\), confirming it acts downward.
Datum choice is yours — pick it to kill the most terms. Putting the datum at the lowest point of travel makes every \(V_g\) positive and removes a common minus-sign slip. Just be consistent across positions 1 and 2.
14.6Conservation of Energy
Splitting work into conservative and non-conservative parts and using \(U_{1\text{-}2,\text{cons}} = V_1 - V_2\), the work–energy principle becomes a balance of kinetic and potential energies.
If only conservative forces do work, the sum of kinetic and potential energies remains constant throughout the motion — kinetic energy converts to potential energy and back.
\[ T_1 + V_1 = T_2 + V_2 \]For a system of particles, \(\sum T_1 + \sum V_1 = \sum T_2 + \sum V_2\) (Eqs. 14–21, 14–22), i.e. \(\sum T + \sum V = \text{const}\).
Procedure for Analysis
- Draw the particle at its initial and final positions; fix a horizontal datum for \(V_g\).
- From the geometry, read off the elevation \(y\) and any spring stretch/compression \(s\).
- Recall \(V_g = Wy\) (sign by datum) and \(V_e = \tfrac{1}{2}ks^2\) (always positive).
- Apply \(T_1 + V_1 = T_2 + V_2\); measure \(v\) from an inertial frame.
Examples 14.9 and 14.11 show the standard combo move: use conservation of energy to get the speed, then the equation of motion along the \(n\)-axis to get the normal force or tension. Energy alone can't give you a force that does no work — you'll always need that second step.
Physical meaning
Work is force accumulated through displacement, whereas kinetic and potential energies describe state. A positive net work raises speed; a negative net work lowers speed, but neither statement supplies motion direction or elapsed time.
Problem-solving strategy
- Choose work–energy when speeds at positions, stopping distance, springs, friction over travel, or power are central; choose impulse–momentum instead when the decisive interval is specified in time.
- Define the particle or particle system, path coordinate, positive direction, energy datum, and spring natural length before writing terms.
- Inventory every force and decide whether to integrate its work, represent it by potential energy, or assign zero work because it is perpendicular to displacement.
- Write the scalar balance symbolically, solve, and reject a negative kinetic-energy result as evidence that the assumed endpoint cannot be reached.
- Audit units, work signs, contact conditions, friction regime, and whether the calculated speed lies within model and equipment limits.
Integrated applications
Recognition strategy
The data connect force to position and ask for speed after a displacement, so work–energy is direct. Time and acceleration history are neither supplied nor required.
Given information
- Particle mass \(m=18.0\ \text{kg}\) and initial speed \(v_1=2.00\ \text{m/s}\).
- Coordinate \(+x\) is right; displacement is from \(x=0\) to \(x=5.00\ \text{m}\).
- Drive force is position-dependent and drag is a constant opposing force; the level-rail normal and weight do zero work.
Model and interpretation
Model the cart as a particle constrained to a horizontal path. The rail remains in contact, drag retains its stated direction, and the drive law is valid throughout the interval. This is kinetics because forces determine the speed change.
Verification
Units: Each force–distance integral has \(\text{N}\!\cdot\!\text{m}\), equivalent to \(\text{J}\), and both kinetic-energy terms have \(\text{kg}\!\cdot\!\text{m}^2/\text{s}^2\), so the equation is dimensionally consistent.
Signs and direction: Drive work is positive and drag work is negative under the declared right-positive convention. Positive net work correctly produces a larger speed magnitude.
Plausibility: The average drive is 70 N, leaving an average net force of 45 N; over 5 m this gives 225 J, exactly matching the integrated result and making 5.39 m/s reasonable.
Engineering interpretation
The result depends on the area under the measured force–position curve, not merely its final value. Rail losses or a drive-law calibration outside this travel range would require revised work data.
Recognition strategy
The target speed is tied to travel, friction, gravity, and changing spring deformation. A single work–energy balance includes all four effects without solving acceleration as a function of position.
Given information
- Take the uphill tangent as positive; \(m=12.0\ \text{kg}\), \(s=2.00\ \text{m}\), and \(v_1=1.50\ \text{m/s}\).
- Actuator force is 500 N uphill; \(\mu_k=0.180\), \(g=9.81\ \text{m/s}^2\), and incline angle is \(20.0^\circ\).
- The linear spring has \(k=250\ \text{N/m}\) and deformation measured from natural length changes from 0.100 m to 2.100 m.
Model and interpretation
Treat the slider as a particle that remains on the rigid incline and is already sliding uphill, so friction acts downhill. The actuator and spring are parallel to the plane, giving \(N=mg\cos20^\circ\); the spring is linear over the stated range.
Verification
Units: Every work contribution is force times distance in joules, while the initial and final kinetic energies use kilograms and squared metres per squared second, also joules.
Signs and direction: Uphill actuator work is positive; gravity, friction, and the increasingly stretched spring oppose uphill displacement and therefore enter negatively. The positive final radicand confirms admissible motion.
Plausibility: Net work is about 330 J and initial kinetic energy is 13.5 J, so final energy near 343 J implies \(v\approx\sqrt{2(343)/12}=7.56\ \text{m/s}\), consistent with the result.
Engineering interpretation
The spring absorbs more than half the actuator work, so stiffness and installed preload strongly affect speed. A real actuator selection must also check stroke, force rating, thermal duty, and the spring linearity limit.
Recognition strategy
The question asks for instantaneous power at a specified speed. First obtain cable force from kinetics, then use the force–velocity power relation and apply efficiency in the correct direction.
Given information
- Module mass is 900 kg; define upward as positive and use \(g=9.81\ \text{m/s}^2\).
- At the instant considered, \(v=1.20\ \text{m/s}\) upward and \(a=0.300\ \text{m/s}^2\) upward.
- Efficiency is \(\eta=P_{\text{cable}}/P_{\text{in}}=0.820\); cable mass and other losses are represented only by this supplied efficiency.
Model and interpretation
Model the module as a particle and the taut cable as applying an upward tension. Newton’s law is needed for the instantaneous force; power then measures its energy-transfer rate. Positive cable power means energy enters the module.
Verification
Units: Tension from kg m/s² is newtons; multiplying by m/s gives N m/s, which is watts. Dividing by dimensionless efficiency preserves power units.
Signs and direction: Tension and velocity point upward, so cable power into the load is positive. Acceleration upward makes tension exceed weight, and an efficiency below one makes input exceed output.
Plausibility: At constant speed the load alone would require \(mgv=10.59\ \text{kW}\) at the cable. Acceleration and losses must raise the demand, so 13.32 kW is physically ordered.
Engineering interpretation
This instantaneous requirement informs motor and drive sizing, but a professional selection must also check startup transients, regenerative lowering, duty cycle, brake capacity, cable factors, and applicable lifting codes.
★Chapter Review
Work of a Force
A force does work over a displacement along its line of action. For a variable force, \(U = \int F\cos\theta\,ds\) — the area under the \(F\)–\(s\) diagram. Constant force: \(U = F_c\,\Delta s\). Weight: \(U = -W\Delta y\). Spring: \(U = -(\tfrac{1}{2}ks_2^2 - \tfrac{1}{2}ks_1^2)\).
Principle of Work and Energy
Combining \(\sum F_t = ma_t\) with \(a_t\,ds = v\,dv\) gives \(T_1 + \sum U_{1\text{-}2} = T_2\). Useful for force–velocity–displacement problems; draw the free-body diagram first.
Power and Efficiency
Power is the time rate of work, \(P = dU/dt = \mathbf{F}\cdot \mathbf{v}\). Efficiency \(\varepsilon = \text{output}/\text{input}\) is always less than one due to friction losses.
Conservation of Energy
Conservative forces (weight, spring) do path-independent work measured by potential energy: \(V_g = \pm Wy\), \(V_e = \tfrac{1}{2}ks^2\). When only conservative forces act, \(T_1 + V_1 = T_2 + V_2\) — mechanical energy is constant.
Decision rule for the exam: friction or non-conservative force present? use work–energy (Eq. 14–7). Only weight and springs? jump straight to conservation of energy (Eq. 14–21) — it needs data at just two points and is faster every time.
Chapter recap
- Select work–energy for displacement-based speed questions and power for instantaneous energy-transfer rate.
- Distinguish work by nonconservative forces from changes in gravitational and elastic potential energy.
- Preserve force direction, spring datum, friction regime, and contact admissibility in every model.
Before moving on
You should be able to construct and verify a complete work–energy balance, integrate variable work, compute power and efficiency, and explain when another kinetics method is needed.