7.4 Power

Power and energy transfer are fundamental concepts in physics, describing how work is done and energy moves between systems. Understanding power calculations and rates helps us analyze everything from car engines to light bulbs, showing how energy changes over time.

In real-world scenarios, power output is crucial for machines, engines, and electrical systems. We'll explore how power relates to force and velocity in moving objects, and see how these concepts apply to everyday situations like pushing boxes or generating electricity.

Power and Energy Transfer

Power calculation and rate

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• Power ($P$) represents the rate at which work is performed or energy is transferred over time
• Mathematically expressed as $P = \frac{W}{\Delta t}$, where $W$ is work done or energy transferred and $\Delta t$ is the time interval
• Measured in watts (W), which is equivalent to joules per second (J/s)
• Can also be calculated using the rate of change of kinetic energy: $P = \frac{\Delta KE}{\Delta t}$
  • Kinetic energy formula: $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity
  • Examples: power output of a car engine (horsepower), energy consumption of a light bulb (watts)

Power concept in moving objects

• Power expressed in terms of force ($F$) and velocity ($v$): $P = Fv$
  • Derived from the work-energy theorem: $W = F \cdot d$, where $d$ is displacement
  • Substituting $d = v \Delta t$ into the work-energy equation and dividing by $\Delta t$ yields $P = Fv$
• When force is applied at an angle $\theta$ relative to the object's motion, power is given by $P = Fv \cos \theta$
  • Accounts for the component of the force in the direction of the object's motion
  • Examples: power generated by a wind turbine, power required to push a box up an incline
• Torque, the rotational equivalent of force, is related to power in rotating systems

Power output in real-world scenarios

• Mechanical power output calculated for machines and engines
  • Car engine producing 150 horsepower (hp) is approximately 112 kilowatts (kW)
    • 1 hp ≈ 745.7 W
  • Power generated by a hydroelectric dam turbine (megawatts)
• Electrical power is the product of voltage ($V$) and current ($I$): $P = VI$
  • Voltage measured in volts (V) and current measured in amperes (A)
  • Expressed in watts (W) or kilowatts (kW)
  • Examples: power consumption of household appliances (refrigerator, television), power output of a solar panel
• When solving power problems, ensure all quantities are expressed in consistent units
  • Convert units as necessary to maintain consistency throughout calculations
  • Examples: converting horsepower to watts, expressing energy in joules instead of kilowatt-hours

Energy Efficiency and Conservation

• Energy efficiency is the ratio of useful energy output to total energy input
• Conservation of energy principle states that energy cannot be created or destroyed, only converted from one form to another
• Power factor in electrical systems affects the efficiency of power transmission
• Energy dissipation occurs when energy is converted to less useful forms, such as heat, during power transfer processes