Unit 3: Work, Energy, and Power - AP Physics 1

1. Translational Kinetic Energy

• Definition: Kinetic energy is defined as the energy of motion, which means any object in motion has kinetic energy.

• Equation: The mathematical formula to calculate kinetic energy is:

  [ KE = \frac{1}{2} m v^2 ]

  • Where:

    • KE = kinetic energy (measured in Joules, J)

    • m = mass of the object (in kilograms, kg)

    • v = speed of the object (in meters per second, m/s)

• Characteristics:

  • Kinetic energy is always a positive value or zero; it cannot be negative.

  • The measurement of kinetic energy is dependent on the frame of reference, meaning that different observers may calculate different values for the same object based on their relative motion.

• Practice Problems:

  • Calculate the kinetic energy of a 4 kg object moving at a speed of 10 m/s.

  • If the speed of the object doubles, how does the kinetic energy change?

  • Answer: If the speed doubles, kinetic energy increases by a factor of four since KE is proportional to the square of the speed.

2. Work

• Definition: Work is defined as the amount of mechanical energy that is transferred into or out of a system through the application of a force over a distance.

• Equation for Work Done by a Constant Force: The general equation to find work is:

  [ W = F \cdot d \cdot \cos(\theta) ]

  • Where:

    • W = work done (in Joules, J)

    • F = force applied (in Newtons, N)

    • d = displacement moved by the object (in meters, m)

    • ( \theta ) = angle between the direction of the applied force and the direction of the displacement.

• Work Characteristics:

• Work done by conservative forces (like gravity) does not depend on the path taken; it only relates to the initial and final states.

• Conversely, work done by nonconservative forces (like friction) is dependent on the path taken.

• Practice Problems:

  • A force of 50 N is applied to move an object 3 meters in the direction of the force. Calculate the work done.

    • Answer: W = 50 N × 3 m = 150 J

  • If the displacement is at an angle of 30 degrees to the direction of the force, how does this affect the work done?

3. Mechanical Energy

• Types of Mechanical Energy:

  • 1. Kinetic Energy: Energy an object has due to its motion.

  • 2. Gravitational Potential Energy: Energy stored in an object due to its height in a gravitational field.

  • Equation:

  [ PE = mgh ]

  • Where:

    • PE = gravitational potential energy (measured in Joules, J)

    • m = mass of the object (kg)

    • g = acceleration due to gravity (approximately 9.8 m/s²)

    • h = height above a reference point (in meters, m)

  • 3. Elastic Potential Energy: Energy stored in elastic materials, such as springs.

  • Equation:

  [ PE_{elastic} = \frac{1}{2} k x^2 ]

  • Where:

    • k = spring constant (N/m)

    • x = displacement from the equilibrium position (m)

• Conservation of Mechanical Energy:In a closed system with only conservative forces at play, the total mechanical energy (the sum of kinetic and potential energy) remains constant.

• Practice Problems:

  • What is the gravitational potential energy of a 5 kg object at a height of 10 meters?

    • Answer: PE = 5 kg × 9.8 m/s² × 10 m = 490 J

  • If an object falls from 20 meters, calculate its potential energy change as it reaches the ground.

    • Answer: Initial PE = mgh = 5 kg × 9.8 m/s² × 20 m = 980 J; final PE at ground = 0 J; change = 980 J.

4. Power

• Definition: Power is defined as the rate at which work is done or energy is transferred over time.

• Equations for Power:

  • Average Power:

    [ P_{avg} = \frac{W}{\Delta t} ]

    • Where:

      • P = power (measured in Watts, W)

      • W = work done (Joules, J)

      • ( \Delta t ) = time interval (seconds, s)

  • Instantaneous Power:

    [ P = F v \cos(\theta) ]

• Units: Power is measured in watts (W), where 1 Watt = 1 Joule/second.

• Practice Problems:

  • If 300 Joules of work are done in 5 seconds, what is the average power?

    • Answer: P = 300 J / 5 s = 60 W

  • A machine does 1500 J of work with a power output of 300 W. How long did it take to do this work?

    • Answer: Time = W / P = 1500 J / 300 W = 5 s

• Importance of Power: Understanding power is crucial in various fields including engineering, electricity, and biomechanics as it helps in designing efficient machines and understanding energy consumption.