Kinetics of Particles: Work and Energy - Theory & Concepts

Kinetics of Particles: Work and Energy

Learning Objectives

Define work, kinetic energy, potential energy, power, and efficiency.
Apply the principle of work and energy to particle kinetics.
Distinguish between conservative and non-conservative forces.
Apply the conservation of mechanical energy principle.

The method of work and energy is a powerful tool for solving problems involving force, displacement, and velocity, without explicitly determining acceleration. It relates the change in kinetic energy of a particle to the work done on it.

Work of a Force

Work is the energy transferred to or from a particle by a force acting through a distance.

Work

Work is defined as the energy transferred to or from a particle by a force acting through a distance.

General Work Equation

The work done by a force moving a particle through a differential displacement.

U = \int \mathbf{F} \cdot d\mathbf{r}

Variables

Symbol	Description	Unit
$U$	Work done	J or ft-lb
$\mathbf{F}$	Force vector	N or lb
$d\mathbf{r}$	Differential displacement vector	m or ft

Work by a Constant Force in a Straight Line

Simplified work equation for a constant force.

U = F \Delta s \cos \theta

Variables

Symbol	Description	Unit
$U$	Work done	J or ft-lb
$F$	Magnitude of constant force	N or lb
$\Delta s$	Linear displacement	m or ft
$\theta$	Angle between force and displacement vectors	rad or deg

Work Done by Gravity (Weight)

Work done by gravity depends only on the vertical displacement.

U_g = -W \Delta y = -mg(y_2 - y_1)

Variables

Symbol	Description	Unit
$U_g$	Work done by gravity (negative if moving up, positive if moving down)	J
$W$	Weight of the particle	N
$m$	Mass	kg
$g$	Acceleration due to gravity	$m/s^2$
$y_1, y_2$	Initial and final vertical positions	m
$\Delta y$	Change in vertical position	m

Work Done by a Spring

Work done by a linear spring when stretched or compressed.

U_s = -\frac{1}{2} k (x_2^2 - x_1^2)

Variables

Symbol	Description	Unit
$U_s$	Work done by a spring (negative when moving away from equilibrium)	J
$k$	Spring stiffness constant	N/m
$x_1, x_2$	Initial and final deformation from unstretched length	m

Energy

Energy is the capacity to do work. In particle kinetics, we focus on kinetic and potential energy.

Kinetic Energy

Energy due to the motion of a particle. It is always positive.

T = \frac{1}{2} m v^2

Variables

Symbol	Description	Unit
$T$	Kinetic energy	J
$m$	Mass	kg
$v$	Velocity	m/s

Potential Energy

Energy due to position (stored work), including gravitational and elastic potential energy.

V_g = mgh \quad \text{and} \quad V_e = \frac{1}{2} k x^2

Variables

Symbol	Description	Unit
$V_g$	Gravitational potential energy (relative to a datum where h=0)	J
$V_e$	Elastic potential energy (always positive)	J
$m$	Mass	kg
$g$	Acceleration due to gravity	$m/s^2$
$h$	Height above datum	m
$k$	Spring stiffness constant	N/m
$x$	Deformation from unstretched length	m

Power and Efficiency

Power ( $P$ ) is defined as the time rate of doing work. It provides a measure of how fast energy is being transferred.

Power Equation

The time rate of doing work.

P = \frac{dU}{dt} = \frac{\mathbf{F} \cdot d\mathbf{r}}{dt} = \mathbf{F} \cdot \mathbf{v}

Variables

Symbol	Description	Unit
$P$	Power	W or hp
$dU$	Differential work	J or ft-lb
$dt$	Differential time	s
$\mathbf{F}$	Applied force vector	N or lb
$d\mathbf{r}$	Differential displacement vector	m or ft
$\mathbf{v}$	Velocity vector	m/s or ft/s

Units of Power

SI: Watts ( $W$ ), where $1 \text{ W} = 1 \text{ J/s} = 1 \text{ N} \cdot \text{m/s}$ .
US Customary: Horsepower ( $hp$ ), where $1 \text{ hp} = 550 \text{ ft} \cdot \text{lb/s} = 746 \text{ W}$ .

The mechanical efficiency ( $\eta$ ) of a machine is the ratio of the useful power produced (power output) to the power supplied to the machine (power input). Because energy is always lost to friction or heat in real machines, efficiency is always less than 1 (or $\lt 100\%$ ).

Mechanical Efficiency

Ratio of output power/work to input power/work.

\eta = \frac{P_{out}}{P_{in}} = \frac{W_{out}}{W_{in}}

Variables

Symbol	Description	Unit
$\eta$	Mechanical efficiency	dimensionless
$P_{out}$	Useful power output	W or hp
$P_{in}$	Power input	W or hp
$W_{out}$	Useful work output	J or ft-lb
$W_{in}$	Work input	J or ft-lb

Principle of Work and Energy

The principle relates the total work done on a particle to the change in its kinetic energy.

The initial kinetic energy ( $T_1$ ) plus the total work done by all forces ( $\sum U_{1-2}$ ) equals the final kinetic energy ( $T_2$ ).

Work and Energy Equation

Relates the total work done on a particle to its change in kinetic energy.

T_1 + \sum U_{1-2} = T_2

Variables

Symbol	Description	Unit
$T_1$	Initial kinetic energy	J or ft-lb
$\sum U_{1-2}$	Total work done by all forces from state 1 to state 2	J or ft-lb
$T_2$	Final kinetic energy	J or ft-lb

Advantages of the Work-Energy Method

This method eliminates the need to solve for acceleration, making it ideal for problems involving forces that vary with position (like springs) or path-dependent problems. However, it cannot directly determine acceleration or time.

Interactive Simulation

Interact with the simulation below to explore work and energy concepts.

Work-Energy & Incline Simulator

Incline: 15° | μ_k: 0.10

Control Parameters

Incline Angle (

\theta

)15°

Mass (m)2.0 kg

Spring stiffness (k)500 N/m

Initial Compression (c)0.20 m

Friction (

\mu_k

)0.10

Work-Energy Conservation

T_1 + V_1 + U_{1 \to 2} = T_2 + V_2

T_1 = 0 \text{ J}

V_1 = PE_{sp, 1} = \frac{1}{2}(500)(0.20)^2 = 10.0 \text{ J}

U_{1 \to 2} = -W_f = -0.0 \text{ J}

E_{total, 1} = 10.0 - 0.0 = 10.0 \text{ J}

T_2 = KE = \frac{1}{2}(2)(0.00)^2 = 0.0 \text{ J}

V_2 = PE_{sp} + PE_g = 10.0 + 0.0 = 10.0 \text{ J}

E_{total, 2} = 0.0 + 10.0 = 10.0 \text{ J}

Position (x):-0.200 m

Velocity (v):0.00 m/s

Height (h):0.00 m

Dynamic Energy Allocation (Joules)

PE Spring10.0J

PE Gravity0.0J

KE (Motion)0.0J

Friction Loss0.0J

Total E10.0J

Conservative vs. Non-Conservative Forces

Force Types

Conservative Forces: The work done by these forces is independent of the path taken; it depends only on the initial and final positions. Gravity and spring forces are conservative. They allow for the definition of potential energy.
Non-Conservative Forces: The work done depends on the path taken. Friction and applied mechanical forces are non-conservative. They dissipate or add energy to the system.

Conservation of Energy

When only conservative forces do work on a system, the work they do can be expressed as a change in potential energy ( $U = -\Delta V$ ). This leads to the conservation of mechanical energy principle.

Conservation of Energy Equation

Total mechanical energy is conserved when only conservative forces do work.

T_1 + V_1 = T_2 + V_2

Variables

Symbol	Description	Unit
$T_1$	Initial kinetic energy	J or ft-lb
$V_1$	Initial potential energy (V = V_g + V_e)	J or ft-lb
$T_2$	Final kinetic energy	J or ft-lb
$V_2$	Final potential energy	J or ft-lb

Non-Conservative Forces

If non-conservative forces do work (such as friction or an applied force), mechanical energy is not conserved. The equation must be modified to include the work of non-conservative forces.

Work-Energy with Non-Conservative Forces

Modified conservation equation including the work of non-conservative forces.

T_1 + V_1 + \sum (U_{NC})_{1-2} = T_2 + V_2

Variables

Symbol	Description	Unit
$T_1$	Initial kinetic energy	J or ft-lb
$V_1$	Initial potential energy	J or ft-lb
$\sum (U_{NC})_{1-2}$	Total work done by non-conservative forces	J or ft-lb
$T_2$	Final kinetic energy	J or ft-lb
$V_2$	Final potential energy	J or ft-lb

Key Takeaways

Principle of Work and Energy ( $T_1 + \Sigma U = T_2$ ) relates speed and displacement. It does not involve time directly.
Kinetic Energy ( $T = \frac{1}{2}mv^2$ ) is scalar and always non-negative.
Conservative Forces (Gravity, Springs) allow work to be expressed as a change in potential energy.
Non-Conservative Forces (Friction, Applied Forces) depend on the path and alter the total mechanical energy of the system.
Work of Friction is always negative because friction opposes motion.
Conservation of Energy ( $T_1 + V_1 = T_2 + V_2$ ) applies only when non-conservative forces do no work.

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