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 UU done by a force F\mathbf{F} moving a particle through a differential displacement drd\mathbf{r} is:
U=∫F⋅drU = \int \mathbf{F} \cdot d\mathbf{r}
For a constant force moving in a straight line:
U=FΔscos⁡θU = F \Delta s \cos \theta
Where θ\theta is the angle between the force vector and displacement vector.

Common Work Expressions

  • Constant Force: U=Fdcos⁡θU = F d \cos \theta
  • Weight (WW): The work done by gravity depends only on the vertical displacement Δy\Delta y. U=−WΔy=−mg(y2−y1)U = -W \Delta y = -mg(y_2 - y_1) (Negative if moving up, positive if moving down).
  • Spring Force (Fs=kxF_s = kx): The work done by a spring is negative when it is stretched or compressed from its equilibrium position. U=−12k(x22−x12)U = -\frac{1}{2} k (x_2^2 - x_1^2) Where xx is the deformation from the unstretched length.

Energy

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

Types of Energy

  1. Kinetic Energy (TT): Energy due to motion. T=12mv2T = \frac{1}{2} m v^2 Note: Kinetic energy is always positive (since v2≥0v^2 \ge 0).
  2. Potential Energy (VV): Energy due to position (stored work).
    • Gravitational Potential Energy: Vg=mghV_g = mgh (relative to a datum where h=0h=0).
    • Elastic Potential Energy: Ve=12kx2V_e = \frac{1}{2} k x^2 (always positive).

Power and Efficiency

Power

Power (PP) is defined as the time rate of doing work. It provides a measure of how fast energy is being transferred.
P=dUdt=F⋅drdt=F⋅vP = \frac{dU}{dt} = \frac{\mathbf{F} \cdot d\mathbf{r}}{dt} = \mathbf{F} \cdot \mathbf{v}
Where:
  • F\mathbf{F} is the applied force.
  • v\mathbf{v} is the velocity of the point of application of the force.
Units:
  • SI: Watts (WW), where 1 W=1 J/s=1 N⋅m/s1 \text{ W} = 1 \text{ J/s} = 1 \text{ N} \cdot \text{m/s}.
  • US Customary: Horsepower (hphp), where 1 hp=550 ft⋅lb/s=746 W1 \text{ hp} = 550 \text{ ft} \cdot \text{lb/s} = 746 \text{ W}.

Mechanical Efficiency

The mechanical efficiency (η\eta) of a machine is the ratio of the useful power produced (power output) to the power supplied to the machine (power input).
η=PoutPin=WoutWin\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 <100%\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

T1+∑U1−2=T2T_1 + \sum U_{1-2} = T_2
The initial kinetic energy (T1T_1) plus the total work done by all forces (∑U1−2\sum U_{1-2}) equals the final kinetic energy (T2T_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.

System Parameters

2.0
500
0.20
0.00
2kg
-0.5m0m0.5m1.0m1.5m2.0m

Energy Conservation (J)

PE0
KE0
Heat0
Total0

Velocity: 0.00 m/s

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=−ΔVU = -\Delta V). This leads to the conservation of mechanical energy principle.

Conservation of Energy Equation

T1+V1=T2+V2T_1 + V_1 = T_2 + V_2
Where V=Vg+VeV = 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 (UNC)(U_{NC}):
T1+V1+∑(UNC)1−2=T2+V2T_1 + V_1 + \sum (U_{NC})_{1-2} = T_2 + V_2
Key Takeaways
  • Principle of Work and Energy (T1+ΣU=T2T_1 + \Sigma U = T_2) relates speed and displacement. It does not involve time directly.
  • Kinetic Energy (T=12mv2T = \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 (T1+V1=T2+V2T_1 + V_1 = T_2 + V_2) applies only when non-conservative forces do no work.
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