Work, Energy, and Power Samar Hathout KDTH 101

Work is the transfer of energy through motion. In order for work to take place, a force must be exerted through a distance. The amount of work done depends on two things: the amount of force exerted and the distance over which the force is applied. There are two factors to keep in mind when deciding when work is being done: something has to move and the motion must be in the direction of the applied force. Work can be calculated by using the following formula: Work=force x distance Work

Work is done on the books when they are being lifted, but no work is done on them when they are being held or carried horizontally. Work

Work can be positive or negative Man does positive work lifting box Man does negative work lowering box Gravity does positive work when box lowers Gravity does negative work when box is raised

Work done by a constant Force   E kin = W net W = F s = |F| |s| cos  = F s s |F| : magnitude of force |s| = s : magnitude of displacement F s = magnitude of force in direction of displacement : F s = |F| cos   : angle between displacement and force vectors Kinetic energy : E kin = 1/2 m v 2 Work-Kinetic Energy Theorem: F s

Work Done by Gravity l Example 1: Drop ball Y i = h Y f = h f W g = (mg)(S) S = h 0 -h f W g = mg(h 0 -h f ) = mg(h 0 -h f ) = E pot,initial – E pot,final mg S y x Y i = h 0 mg S y x

Work Done by Gravity Example 2: Toss ball up W g = (mg)(S) S = h 0 -h f W g =-mg(h 0 -h f ) = E pot,initial – E pot,final Y i = h 0 Y f = h f mg S y x

Work Done by Gravity Example 3: Slide block down incline W g = (mg)(S)cos  S = h/cos  W g = mg(h/cos  )cos  W g = mgh with h= h 0 -h f h  mg S  Work done by gravity is independent of path taken between h 0 and h f => The gravitational force is a conservative force. h0h0 h f

Work done by a Variable Force l The magnitude of the force now depends on the displacement: F s (s) Then the work done by this force is equal to the area under the graph of F s versus s, which can be approximated as follows: W =  W i  F s (s i )  s = (F s (s 1 )+F s (s 2 )+…)  s

Concept Question Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball reach the bottom with the highest speed? 1. Dropping 2. Slide on ramp (no friction) 3. Swinging down 4. All the same In all three experiments, the balls fall from the same height and therefore the same amount of their gravitational potential energy is converted to kinetic energy. If their kinetic energies are all the same, and their masses are the same, the balls must all have the same speed at the end correct

Kinetic EnergyPotential Energy Types of Energy

Radiant Electrical Chemical Thermal Nuclear Magnetic Sound Mechanical Forms of Energy

Mechanical energy is the movement of machine parts. Mechanical energy is also the total amount of kinetic and potential energy in a system. Wind-up toys, grandfather clocks, and pogo sticks are examples of mechanical energy. Wind power uses mechanical energy to help create electricity. Potential energy + Kinetic energy = Mechanical energy Mechanical Energy

Potential energy + Kinetic energy = Mechanical energy Example of energy changes in a swing or pendulum. Mechanical Energy

Conservation of Mechanical Energy l Total mechanical energy of an object remains constant provided the net work done by non-conservative forces is zero: E tot = E kin + E pot = constant or E kin,f +E pot,f = E kin,0 +E pot,0 Otherwise, in the presence of net work done by non-conservative forces (e.g. friction): W nc = E kin,f – E kin,0 + E pot,f -E pot,i

Example Problem Suppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces? 1. 0J 2. 50J J J J correct Work done by non-conservative forces equals the difference between final and initial kinetic energies plus the difference between the final and initial gravitational potential energies. W = (300-75) + ((-25) - 250) = = -50J. Samar Hathout

Kinetic Energy Same units as work Remember the Eq. of motion Multiply both sides by m, Samar Hathout

Example

Potential Energy l Potential energy exists whenever an object which has mass has a position within a force field (gravitational, magnetic, electrical). çWe will focus primarily on gravitational potential energy (energy an object has because of its height above the Earth)

Potential Energy If force depends on distance, For gravity (near Earth’s surface) Samar Hathout

Conservation of Energy Conservative forces: Gravity, electrical, QCD… Non-conservative forces: Friction, air resistance… Non-conservative forces still conserve energy! Energy just transfers to thermal energy Samar Hathout

Example A diver of mass m drops from a board 10.0 m above the water surface, as in the Figure. Find his speed 5.00 m above the water surface. Neglect air resistance. 9.9 m/s

Example A skier slides down the frictionless slope as shown. What is the skier’s speed at the bottom? H=40 m L=250 m start finish 28.0 m/s

Example Three identical balls are thrown from the top of a building with the same initial speed. Initially, Ball 1 moves horizontally. Ball 2 moves upward. Ball 3 moves downward. Neglecting air resistance, which ball has the fastest speed when it hits the ground? A) Ball 1 B) Ball 2 C) Ball 3 D) All have the same speed.

Springs (Hooke’s Law) Proportional to displacement from equilibrium

Potential Energy of Spring  PE=-F  x xx F

x Example b) To what height h does the block rise when moving up the incline? A 0.50-kg block rests on a horizontal, frictionless surface as in the figure; it is pressed against a light spring having a spring constant of k = 800 N/m, with an initial compression of 2.0 cm. 3.2 cm

Power l Average power is the average rate at which a net force does work: P av = W net / t SI unit: [P] = J/s = watt (W) Or P av = F net s /t = F net v av

Example A 1967 Corvette has a weight of 3020 lbs. The 427 cu-in engine was rated at 435 hp at 5400 rpm. a) If the engine used all 435 hp at 100% efficiency during acceleration, what speed would the car attain after 6 seconds? b) What is the average acceleration? (in “g”s) a) 120 mph b) 0.91g

Example Consider the Corvette (w=3020 lbs) having constant acceleration of a=0.91g a) What is the power when v=10 mph? b) What is the power output when v=100 mph? a) 73.1 hp b) 732 hp (in real world a is larger at low v)