Work and Energy

Work

  • W = Fdcosθ
  • F is force, d is the distance over which the force is applied, and θ is the angle between the force and distance.
  • Derived units, sign conventions
    • Work is energy, and the unit is the Joule.
    • Joule = N·m = kg·m/s2·m = kg·m2/s2
    • If the force and the distance applied is in the same direction, work is positive.
    • For example, pushing a crate across a rough terrain involves you doing positive work (you are pushing forward and the crate is moving forward).
    • If the force and the distance applied is in opposite directions, work is negative.
    • For a non-rotating system, friction always does negative work because it acts against the direction of motion.
    • If the force is acting in one direction, but the object moves in a perpendicular direction, then no work is done.
    • The classic example is that no work is done by your arms when you carry a bucket of water for a mile. Because you are lifting the bucket vertically while its motion is horizontal.
    • If you like math, then everything you need to know is already contained in the mathematical formula. Cosine of 90 is zero; cosine of anything below 90 is positive and between 90-180 is negative ...so forth.
  • Amount of work done in gravitational field is path-independent
    • Unlike friction, gravity always acts downwards. Thus, it does not matter what detour you take because sideward motion perpendicular to the gravitational force involves no work.
    • Pushing an object at constant speed up a frictionless inclined plane involves the same amount of work as directly lifting the same object to the same height at constant speed.
    • Sliding down a frictionless inclined plane involves the same gravitational work as doing a free fall at the same height.
  • Mechanical advantage
    • Mechanical advantage = little input force (effort) -> large output force.
    • Using the lever arm can achieve mechanical advantage.
    • Using pulleys can achieve mechanical advantage.
  • Work-kinetic energy theorem
    • Work on an object can transform into kinetic energy.
      • When you pushing on an object, it will move: Fd = ½mv2
      • When gravity does work on an object, it will move: Fweighth = mgh = ½mv2
    • Kinetic energy of an object can do work.
      • A moving object can slide up an inclined plane before coming to a stop: ½mv2 = mgh
      • A moving object can slide against friction for a while before coming to a stop: ½mv2 = Ffrictiond
  • Power
    • Power is the rate of work, or work over time: P = W/t
    • The unit for power is the Watt, or W (don't confuse this W with the shorthand of work).
    • Watt = Joule / second

Energy

  • Work and energy are interchangeable.
  • All types of energy have the same unit - the Joule.
  • Kinetic energy: KE = 1/2 mv^2; units
    • KE = ½mv2
    • Unit = Joule = kg·m2/s2
    • At the same speed, the larger mass has the larger kinetic energy.
    • When you double the mass, you double the kinetic energy.
    • At the same mass, the higher speed has the larger kinetic energy.
    • When you double the speed, you quadruple the kinetic energy.
    • Speed is more important than mass for the kinetic energy because speed is squared.
  • Potential energy
    • PE = mgh (gravitational, local)
      • PE = mgh is local because it only works on the surface of the Earth.
      • h is the distance from the Earth's surface.
      • PE = mgh is derived from a more general formula.
      • On earth, g is 9.8. g is larger for planets with a higher mass to radius ratio.
    • PE = 1/2kx^2 (spring)
      • x is distance of the end of the spring from its equilibrium position.
      • k is the spring constant.
      • Stiff springs have a larger k because they are harder to stretch (it takes more energy to stretch them).
    • PE = -GmM/r (gravitational, general)
      • This is the general formula for gravitational potential energy.
      • r is the distance between the center of the two attracting objects.
      • G is the universal gravitation constant - it is the same for everything.
      • m and M are the mass of the two attracting objects.
  • Conservation of energy
    • The total amount of energy before = the total amount of energy after.
    • Gravitational potential energy is converted to kinetic energy as an object falls, but the total amount of energy stays the same.
    • Kinetic energy is converted to heat and sound energy as a crate slides to a stop on a rough surface.
  • Conservative forces
    • If a force doesn't dissipate heat, sound or light, then it is a conservative force.
    • Work done by conservative forces are path independent.
    • Conservative forces are associated with a potential energy.
    • For example, the force from a spring can be stored as spring potential energy.
    • Gravitational force can be stored as gravitational potential energy.
    • Electromagnetic forces are also conservative.
    • non-conservative include frictional forces and human exertion. When friction acts on an object, the heat and sound released cannot be recovered. When you flex your arm, you lose heat that cannot be recovered (you cannot re-absorb the heat you lost).
  • Power, units
    • Power is the rate of energy use.
    • The unit for power is the Watt, or Joule per second.
    • Lifting a crate in one minute requires more power than lifting the same crate in an hour.