Kinetics of Particles: Work and Energy - Theory & Concepts

Kinetics of Particles: Work and Energy

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 Formula

The work $U$ done by a force $\mathbf{F}$ moving a particle through a differential displacement $d\mathbf{r}$ is:

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

For a constant force moving in a straight line:

U = F \Delta s \cos \theta

Where $\theta$ is the angle between the force vector and displacement vector.

Common Work Expressions

Constant Force: $U = F d \cos \theta$
Weight ( $W$ ): The work done by gravity depends only on the vertical displacement $\Delta y$ . $U = -W \Delta y = -mg(y_2 - y_1)$ (Negative if moving up, positive if moving down).
Spring Force ( $F_s = kx$ ): The work done by a spring is negative when it is stretched or compressed from its equilibrium position. $U = -\frac{1}{2} k (x_2^2 - x_1^2)$ Where $x$ is the deformation from the unstretched length.

Energy

Energy is the capacity to do work. In particle kinetics, we focus on:

Types of Energy

Kinetic Energy ( $T$ ): Energy due to motion. $T = \frac{1}{2} m v^2$ Note: Kinetic energy is always positive (since $v^2 \ge 0$ ).
Potential Energy ( $V$ ): Energy due to position (stored work).
- Gravitational Potential Energy: $V_g = mgh$ (relative to a datum where $h=0$ ).
- Elastic Potential Energy: $V_e = \frac{1}{2} k x^2$ (always positive).

Power and Efficiency

Power

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

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

Where:

$\mathbf{F}$ is the applied force.
$\mathbf{v}$ is the velocity of the point of application of the force.

Units:

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

Mechanical Efficiency

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).

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

Because energy is always lost to friction or heat in real machines, efficiency is always less than 1 (or $\lt 100\%$ ).

Principle of Work and Energy

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

Work and Energy Equation

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

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

Note

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.

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

T_1 + V_1 = T_2 + V_2

Where $V = V_g + V_e$ .

Important

If non-conservative forces do work (such as friction or an applied force), energy is not conserved. The equation is modified to include the work of non-conservative forces $(U_{NC})$ :

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

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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