Work, Power, and Energy

• Covers work, power, and energy.

Work

• Work is represented as $\tau$.

Work of a Constant Force

• Defined by the formula $\tau = F \cdot d \cdot \cos{\theta}$
  • $\tau$ is work.
  • F is force.
  • d is displacement.
  • $\theta$ is the angle between force and displacement.
• Key observations:
  • If $\theta$ = 0, then $\tau$ = F.d, representing maximum work.
  • If $\theta$ = 90°, then $\tau$ = 0, indicating no work is done.
  • If 0 ≤ $\theta$ < 90°, then $\tau$ > 0, which is work done by the system.
  • If 90° < $\theta$ ≤ 180°, then $\tau$ < 0, representing work done on the system.
• Total work is the sum of individual works: $\tau_{total}$ = $\tau_1 + \tau_2 +... + \tau_n$

Work of a Variable Force

• Equivalent to the area under the curve of a Force vs. displacement (F x d) graph.

Power

Average Power

• Expressed as $P_m = \frac{\tau}{\Delta t}$
  • $P_m$ is average power.
  • $\tau$ is work.
  • $\Delta t$ is the time interval.
• $P_m$ can also be calculated as $F \cdot v_m$, where $v_m$ is average velocity.

Instantaneous Power

• Defined as the limit of average power as the time interval approaches zero: $P = \lim_{\Delta t \to 0} \frac{\tau}{\Delta t}$

Energy

Kinetic Energy

• Defined as $E_c = \frac{m \cdot v^2}{2}$
  • $E_c$ is kinetic energy.
  • m is mass.
  • v is velocity.

Kinetic Energy Theorem (TEC)

• Total work equals the change in kinetic energy: $\tau_{total} = \Delta E_c = E_{cf} - E_{ci}$

Potential Energy

Gravitational Potential Energy

• Given by $E_{pg} = m \cdot g \cdot h$
  • $E_{pg}$ is gravitational potential energy.
  • m is mass.
  • g is the acceleration due to gravity.
  • h is height.

Elastic Potential Energy

• Expressed as $E_{pe} = \frac{k \cdot x^2}{2}$
  • $E_{pe}$ is elastic potential energy.
  • k is the spring constant.
  • x is the deformation of the spring.

Mechanical Energy

• Total mechanical energy is the sum of kinetic and potential energies: $E_m = E_c + E_{pg} + E_{pe}$

Conservative System

• The initial mechanical energy equals the final mechanical energy: $E_{mi} = E_{mf}$

Non-Conservative System

• The change in mechanical energy equals the work done by non-conservative forces: $\Delta E_m = \tau_{Fnc}$ $\tau_{Fnc}$ represents the work done by non-conservative forces.