Chapter 5 – Work and Energy If an object is moved by a force and the force and displacement are in the same direction, then work equals the product of the magnitudes of the force and displacement. W = Fd Force is in Newtons, displacement is in meters and work is in Newton*meters which are Joules (J).

Chapter 5 – Work and Energy If there is a force in a direction that makes an angle, θ, with the direction of the displacement, then W = Fd(cos θ) θ is the angle between the directions of the force and the displacement.

Chapter 5 – Work and Energy Kinetic Energy is the energy of an object in motion. K.E. = ½ mv 2 This formula is derived from a motion formula and the work formula. Kinetic energy depends on an object’s mass and velocity.

Chapter 5 – Work and Energy Work-Kinetic Energy Theorem The net work done on an object is equal to the change in the kinetic energy of the object. W net = ΔK.E. = ½ mv f 2 - ½ mv i 2 = F net d

Positive or Negative Work? Force in direction of motion  speed increases  positive work. Force opposes motion  speed decreases  negative work Force is 90º to motion  no work done Object is not in motion  no work done

Work-kinetic energy theorem examples A car, starting from rest, is pushed with a 253-N force along a level surface. The friction force between the car and the surface is 120 N. How far must the car be pushed so that its final kinetic energy is 2.7 x 10 3 Joules? A 3500-kg car accelerates from rest. The forward force is 781 N and the friction force is 520 N. How far must the car travel to reach a speed of 2.1 m/s? A 2100-kg car starts to roll from the top of a driveway 25 m long that is sloped at angle of 17°. An average friction force of 720 N works against the car’s motion. What will be the car’s speed when it reaches the bottom of the driveway?

Potential Energy Potential energy is the energy associated with an object due to its position (p. 177). Gravitational potential energy is due to the position of an object relative to the Earth or some other gravitational source (p. 177). PE g = mgh (mass x gravity acceleration x height) Units for PE are Joules

Potential Energy (continued) Elastic potential energy is the potential energy in a stretched or compressed elastic object (p. 178) PE elastic = ½ kx 2 k=spring constant (N/m) x=distance stretched or compresed (m) (again, units for PE are Joules)

Potential Energy Examples A 4.5-kg weight is lifted 71 cm above the floor. What is the gravitational potential energy of the weight relative to the floor? A spring with a force constant of 0.83 N/cm has a relaxed length of 1.0 cm. When a mass is attached to the end of the spring, the vertical length becomes 3.1 cm. Calculate the elastic potential energy in stored in the spring.

Conservation of Energy Mass is an example of a conserved quantity m i = m f Mechanical energy is the sum of an object’s kinetic energy and all forms of its potential energy. M.E. = K.E. + P.E. There are other forms of energy that are nonmechanical (like chemical, nuclear, heat, etc.), but for now, we will only deal with mechanical energy.

Conservation of Energy In the absence of friction, mechanical energy is conserved. M.E. i = M.E. f K.E. i + P.E. i = K.E. f + P.E. f

Conservation of mechanical energy examples A flowerpot with a mass of 3.2 kg sits on a window sill, 24 m above the sidewalk. If the flowerpot is pushed off the window sill, what will be its velocity when it hits the sidewalk? A swimmer that weighs 781 N drops off of a board that is 12.0 m above the water’s surface. What is the diver’s velocity when he is 6.0 m above the surface? What will be his velocity when he hits the water’s surface? Suppose the diver in the above problem leaps off the board with an initial speed of 2.3 m/s up. What will his velocity be when he hits the water?

Power Power is the rate at which energy is tranferred (p. 187) P = W / Δ t Power = work ÷ time The unit for power is the Watt (One horsepower, or hp, equals 746 Watts)

Another formula for Power…. Since W = Fd, P = Fd / t Or P = Fv Power = Force x velocity

Power Examples…. A crane at a construction site must lift steel beams to be put in place for the frame of a building. The crane can lift two beams at a time that have a mass of 1200 kg each. What minimum power must the motor in the crane have to lift the beams at a constant speed of 2.70 m/s? What is the power in horsepower (hp)? How long does it take a 15 kW steam engine to do 3.1 x 10 8 J of work? What about a 21 kW steam engine?