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## Work, Power, and Energy

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**Energy, Work, and Power**• Energy • Work • Power • Conservation of Energy**Energy**• The capacity to do work • Easy forms of mechanical energy • Kinetic Energy • Potential Energy • Examples of other types • Radiant • Thermal • Internal • Unit: Joules (J) – one newton x 1 meter**Kinetic Energy**• Definition: Energy due to Motion • KE = (1/2)mv2 m = mass of the object in motion v = velocity of the object Example #1: 10 m/s KE = (1/2)1000*(10)2 = 50,000 J 1000 kg Example #2: 20 m/s KE = (1/2)1000*(20)2 = 200,000 J 1000 kg**Potential Energy**• Definition: Stored Energy due to Position • Gravitational potential energy ~energy due to position PE = mgh • m = mass of object • g = accel.due to gravity (10 m/s2) • h = height of the object (above some arbitrary point) h**Example:**What is the potential energy? m g h = (1kg)(10m/s2)(2m)= 20 Joules Example of Potential Energy 1kg 2 m Don’t worry– it will always be clear what the calculation should be relative to!**Where does the skier have more PE?**When he’s higher! Potential Energy Greater height, Greater mgh, Greater PE Lower height, Smaller mgh, Smaller PE**Springs**• Springs can be compressed or stretched • They have an equilibrium position: the rest position when no forces are applied • x is the distance from equilibrium**Spring Constant**K = Spring constant • Spring constant (k): An indication of how hard it is to stretch (or compress) the spring • Unit: N/m (correct the unit sheet!) • Fs= kx (Hooke’s Law) • Note: Fs is the force required to stretch the spring x meters-- but also the force the spring exerts when it’s stretched that far x Fs**Spring Constant**K = Spring constant • Fs= kx • Example: 15N = k * 0.5m • K = 30 N/m 0.5m 15N**Finding the Spring Constant**• If you’re given force required to compress a spring a distance x, just solve Fs= kx • If you’re given a graph of force vs. distance, just take the slope (or take any point on the graph)**Spring PE**K = Spring constant • The elastic PE or spring PE is given by the equation PEs= ½ kx2 • Example: PEs= ½ 30 * 0.52 = 3.75 J 0.5m 15N**Energy, Work, and Power**• Energy • Work • Power • Conservation of Energy**Work**So we can “pump energy into a system” by doing work on it That increases the total energy When no work is being done, energy is conserved Work and Energy • Work is a means of putting energy into a system (or taking it out) Energy in the system**Work**• Work is a means of putting energy into a system (or taking it out) • Work is done only if a force F causes an object to move a distance D in the direction of the force. W = F d • The units for work are Newton-meters (Nm) or Joules (J). 1 Nm = 1 J**Work**• Remember that d = vt, so if they give you force, velocity and time, you can still find the work • Example: As I push a ball with a force of 10 N, it moves at 3 m/s for 2 s. What is the work done? • Answer: d = vt = 3m/s * 2s = 6 m. W = F d = 10N * 6m = 60 J**Work is done**No work is done Negative work is done– energy is taken out of the system (think of friction or stopping forces) Work • Work is only done by the component of force in the direction of movement. d F F d d F**No work is done by perpendicular component**Work is done by the force in the same direction as the displacement Work • Work is only done by the component of force in the direction of movement. F Fy Fx d W = Fx d**Examples**• I roll my garbage can 20 meters out to the curb with a horizontal force of 100N. • Answer: Force and distance are in same direction. Work = F d = 100N * 20m = 2000 J. • I hold a 10,000N car over my head while rollerskating • Answer: No work is done! My force is up– by displacement is horizontal.**How much work is done lifting a 3-kg box 2m?**How much potential energy was gained? PE vs. Work d = 2m h = 2m F = mg = 30N Work = Fxd = 60 J PE = mgh = 60 J The energy of the box increased by the amount of work done on the box!**How to think about work & energy**• Work = change in energy • When no work is being done, total energy is conserved**Work, Power, and Energy**• Energy • Work • Power • Conservation of Energy**Power**• James Watt, an Englishman in the 1700’s, needed a term to distinguish a “good” steam engine from a “bad” engine. • The difference he determined was the time to do work. • POWER is the rate of doing work. P = W/t Power = Work /t • The units for power are J/s or watts (W) 1 J/s = 1 W**Example Problems**• You exert a 100 N force to move a box 3 m in 4 seconds. How much work did you do on the box? What was your Power Output? 100 N Box 3 meters in 4 seconds W=300 J P=75 W**Another way of looking at power**• Power = W/t = (Fd)/t = F (d/t) = F v • Power = force * velocity • You exert a 100 N force to move a box 3 m in 4 seconds. What was your Power Output? 100 N Box 3 meters in 4 seconds, so v = 0.75 P=100 * 0.75 = 75 W**Example**• An electric pump lifts 5000 N of water from the bottom of a 6 m deep well in 1 minute. • W=F x d = 5,000N x 6m = 30,000 J P= W/t = 30,000J/60s = 500 W**Work, Power, and Energy**• Energy • Work • Power • Conservation of Energy**Conservation of Energy**• Total mechanical energy doesn’t change • …unless we have friction, air resistance, or put work into the system. MEinitial = MEfinal So KEinitial + PEinitial = KEfinal + PEfinal**Diving Board example**PE = 500JKE = 0 • On the trip down, energy is conserved • Total energy is 500J at top • Total energy is 500J all the way down PE = 300JKE = 200J PE = 100JKE = 400J**Diving Board example**PE = 500JKE = 0 • On the trip down, energy is conserved • How did that energy get there? • Work was done! (He climbed up, putting energy into the system) PE = 300JKE = 200J PE = 100JKE = 400J**Work**So we can “pump energy into a system” by doing work on it That increases the total energy When no work is being done, energy is conserved Work and Energy • Work is a means of putting energy into a system (or taking it out) Energy in the system**Work = change in energy**• Equation: W = Fd = ΔET**Diving Board example**PE = 500JKE = 0 • He climbed up, putting energy into the system • How much work did he do? • W = Fd = ΔET • Since he started with 0J, ended with 500J, work= ΔET = 500J PE = 300JKE = 200J PE = 100JKE = 400J**Diving Board example**• Let’s not waste his energy! • He will now do work on the seesaw • How much energy does the diver lose? • W = ΔET = 500J • Which goes into the energy of the seesaw (and then the clown) PE = 500JKE = 0 PE = 300JKE = 200J PE = 100JKE = 400J**Diving Board example**• So 500J of work is done on the clown • W = ΔET = 500J • So the clown’s total energy is now increased (from 0) to around 500J • In real life, some of that 500J would get converted to internal energy (heat, friction, etc.) PE = 500JKE = 0 PE = 300JKE = 200J PE = 0JKE = 0J**We directly model the impact of the force on the object**using PE These forces do work on the object, changing the total mechanical energy When is Energy Conserved? • Not Conserved • Applied forces • Normal forces • Friction • Normal forces • Conserved • Gravitational force • Spring force**Unconserved Situations**• I throw a snowball toward a wall • It has kinetic energy • The wall exerts a normal force on the snowball, removing its kinetic energy • The snowball’s energy is not conserved • (But it is converted to other forms of energy, like internal energy of the wall)**Unconserved Situations, II**• A ball rolls on a surface with friction • It has kinetic energy • The kinetic energy decreases • Friction has done work on the ball to decrease its total energy**Examples…**• I throw a 2-kg ball upward with a velocity of 20m/s. How high does it go? • We’re talking about velocity (KE), and height (PE), so it’s a conservation question • At the beginning (call it h=0): KE = 1/2mv2 = 400J • PE = 0J • Total Energy = 400J • At the top: Total Energy = 400J • KE = 0J (it stops) • PE = 400 – 0 = 400 J • So PE = mgh 400J = 2 * 10 * h • h = 20 m**Example II**• A 1-kg ball starts rolling with a velocity of 10 m/s. A few seconds later, it is rolling at 4 m/s. How much work was done on the ball by friction? • Starting KE = 1/2mv2 = 50 J • Final KE = 8 J • Work = ΔE = 42 J**How to think about work & energy**• Work = change in energy • When no work is being done, total energy is conserved