Unit 3: Work, Energy, and Power

Work and Energy

$$\Delta E = W = F_d d = Fd \cos\theta$$

• In terms of an object or a group of objects (the system), the change in energy of the system equals the work done on the system.
• Work is calculated as the force applied times displacement times the angle between the force and displacement.
• Work causes a change in energy of the system.
• Force parallel to displacement: $$F_{\parallel} = F \cos\theta$$
• Identify which force you are using in the work equation.
• Use the magnitude of the force and the displacement.
• Units for Work: Joules (J) or Newton meters (N·m).

Types of Mechanical Energy

• Kinetic Energy (KE): $$KE = K = \frac{1}{2} mv^2$$ (Cannot be negative).
• Units for energy: $$KE = \frac{1}{2} mv^2 \Rightarrow \left( kg \right) \left( \frac{m}{s} \right)^2 = kg \cdot \frac{m^2}{s^2} = (kg \cdot m/s^2) \cdot m = N \cdot m = J$$
• Elastic Potential Energy (PE): $$PE = U_s = \frac{1}{2} kx^2$$ (Cannot be negative).
• Gravitational Potential Energy (PE): $$PE_g = mgh$$ or $$\Delta U_g = mg\Delta y$$
  • $PE_g$ can be negative. If the object is below the horizontal zero line, then $h$, the vertical height above the zero, is negative.

Work and Energy Properties

• Work and Energy are scalars.
• Conservation of Mechanical Energy: $$ME_i = ME_f$$
  • Valid when there is no energy converted to heat, light, or sound due to friction.
  • Identify the initial and final points.
  • Identify the horizontal zero line.
  • Substitute in mechanical energy values present.
• If there is friction and you need to use energy: $$W_f = \Delta ME$$ (This does not work when there is a force applied).

Power

The rate at which work is done or energy is transferred into or out of the system.

$$P = \frac{\Delta E}{\Delta t} = \frac{W}{\Delta t} = \frac{Fd \cos\theta}{\Delta t} = F v \cos\theta$$

• Units for Power: Watts (W), where 1 Watt = 1 Joule per second.
• $$P = \frac{\Delta E}{\Delta t} = \frac{J}{s} = W$$
• 1 horsepower ($hp$) = 746 watts.

Hooke’s Law

$$|F_s| = k|x|$$

• The force of a spring is linearly proportional to the displacement from its equilibrium position.
• The slope of a Force vs. Displacement graph represents the spring constant.
• $$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{F}{x} = k$$
• Units for the spring constant ($k$) are Newtons per meter: $$\frac{F}{x} = k \Rightarrow \frac{N}{m}$$