Power

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Power is defined as the rate at which __work__ is done over a change in time, but it's also the rate at which energy is transferred over a change in time, where

 AND

Power's variable is the capital letter P, where it's SI unit is in watts (W), after James Watt, and it's U.S. customary system units is in horsepowers (hp), where 1 hp = 550 ft•lb/s = 746 W. But it's also used in the SI units of Joules/second (J/s) because it's quantity is energy (//E//), or __work__ (//W//),over a change in time, and could therefore also be expressed in the SI units of kg•m²/s³. When asking yourself " What is the unit of power ," remember: " Watt is the unit of power ." Power is delivered through many different actions and in many different ways. It can be delivered by any living creature through actions such as sprinting or jumping, and can also be delivered by non-living things like motors that run __cars__, boats, planes, and even elevators, as long as there's some form energy powering some type of matter, there's power. So power is outputted whenever and wherever work is done, or energy is transferred, during a certain amount of time.

Average Power and Instantaneous Power
Average power, the average amount of power outputted during a range of time, and instantaneous power, the amount of power outputted at a specific instance, can be calculated as follows: 

Power Delivered by an Elevator Motor


Problem: An elevator that can carry a maximum of 7.00•10² kg has a mass of 0.50•10³ kg. It's upward motion is acted upon by a constant frictional force of 3.25•10³ N. If the motor of the elevator were to lift the elevator at it's maximum load capacity at a constant velocity of 5.00 m/s, what minimum power should the elevator's motor deliver?

<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> Answers are below in the Sample Problem Solutions section.

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Speedboat Power
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The lake water exerts a constant drag force of magnitude //f// = 4.15•10² N, while Batman's speedboat rests on it's surface. If his boat had a mass of 2.00•10³ kg, what average power would his boat require to be able to speed up to 35.0 m/s in 7.00 s, if the acceleration was constant?

<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answers are below in the Sample Problem Solutions section.

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Power Delivered by the Transfer of an Energy Blast
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Bear with this for just a moment now: if this was possible, just IF this was possible, what would the the average power output of an energy blast be if someone was capable of discharging their own physical energy with an acting force of 4521 N towards a target positioned 2.00 m away from them over the short instance of only 1.00 s?

<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answers are below in the Sample Problem Solutions section.

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Power in a Vertical Jump
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">If a 70.0-kg Austin Powers jumps vertically up with a center of mass velocity at 2.5 m/s, for an instance of only 0.20 s, how much mechanical power did Powers produce? <span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Answers are below in the Sample Problem Solutions section.

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Power Delivered by an Elevator Motor
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">An elevator that can carry a maximum of 7.00•10² kg has a mass of 0.50•10³ kg. It's upward motion is acted upon by a constant frictional force of 3.25•10³ N. If the motor of the elevator were to lift the elevator at it's maximum load capacity at a constant velocity of 5.00 m/s, what minimum power should the elevator's motor deliver? <span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use Newton's second law of motion: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?%5Csum%20%5Cvec%7B%5Cmathbf%7BF%7D%7D=m%5Cvec%7B%5Cmathbf%7Ba%7D%7D]] ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The acceleration is zero because the velocity is constant, so the only forces acting on the elevator are the tension in the cable **T**, the friction **f**, and gravity //M//**g**. (//M// is the mass of the elevator): || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?%5Cvec%7B%5Cmathbf%7BT%7D%7D+%5Cvec%7B%5Cmathbf%7Bf%7D%7D+M%5Cvec%7B%5Cmathbf%7Bg%7D%7D=0]] ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Write the equation in terms of it's components: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?T-f-Mg=0]] ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Solve for //T// and evaluate it: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?T%20=%20f%20+%20Mg]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Substitute //T// in place of the //F// in the power equation: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?P%20=%20Fv%20=%20Tv]]

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Speedboat Power
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The lake water exerts a constant drag force of magnitude //f// = 4.15•10² N, while Batman's speedboat rests on it's surface. If his boat had a mass of 2.00•10³ kg, what average power would his boat require to be able to speed up to 35.0 m/s in 7.00 s, if the acceleration was constant? <span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the work-energy theorem: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?W_%7Bnet%7D=%5CDelta%20K=%5Cfrac%7B1%7D%7B2%7Dmv%5E%7B2%7D_%7Bf%7D%E2%80%93%5Cfrac%7B1%7D%7B2%7Dmv_%7Bi%7D%5E%7B2%7D]] ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Fill in the two work terms and take //vi// = 0: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?W_%7Bengine%7D+W_%7Bdrag%7D=%5Cfrac%7B1%7D%7B2%7Dmv%5E%7B2%7D_%7Bf%7D]] ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the velocity equation of kinematics to find the acceleration: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?v_%7Bf%7D%20=%20at%20+%20v_%7Bi%7D%5Crightarrow%20v_%7Bf%7D=at]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Solve for Δ//x// by substituting a into the time-independent kinematics equation: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?v_%7Bf%7D%5E%7B2%7D-v_%7Bi%7D%5E%7B2%7D=2a%5CDelta%20x]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Find the mechanical energy lost as a result of the drag force by using Δ//x//: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?W_%7Bdrag%7D=%20-f_%7Bd%7D%5CDelta%20x]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Solve for //W//engine: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?W_%7Bengine%7D=%5Cfrac%7B1%7D%7B2%7Dmv_%7Bf%7D%5E%7B2%7D-W_%7Bdrag%7D]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Substitute values into the power formula relevant to work and evaluate: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?P=%5Cfrac%7BW_%7Bengine%7D%7D%7B%5CDelta%20t%7D]]

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Power Delivered by the Transfer of an Energy Blast
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Bear with this for just a moment now: if this was possible, just IF this was possible, what would the the average power output of an energy blast be if someone was capable of discharging their own physical energy with an acting force of 4521 N towards a target positioned 2.00 m away from them over the short instance of only 1.00 s?

<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: || ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the formula for work and substitute the given values into the formula to solve for //W//. || [[image:http://latex.codecogs.com/gif.latex?W%20=%20%5Csum%20F%5CDelta%20xcos%5CTheta]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the formula for power that's relevant to work and substitute in the values to evaluate for //P//. || [[image:http://latex.codecogs.com/gif.latex?P=%5Cfrac%7BW%7D%7B%5CDelta%20t%7D=%5Cfrac%7BW%7D%7Bt_%7Bf%7D-t_%7Bi%7D%7D]]

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Power in a Vertical Jump
<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Problem: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">If a 70.0-kg Austin Powers jumps vertically up with a center of mass velocity at 2.5 m/s, for an instance of only 0.20 s, how much mechanical power did Powers produce?

<span style="color: #0000ff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Solution: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the formula for Kinetic Energy to substitute in the values to evaluate for //K//: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?K=%5Cfrac%7B1%7D%7B2%7Dmv%5E2]]
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Use the formula for power that's relevant to Kinetic Energy to substitute in the values and solve for //P//: || <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">[[image:http://latex.codecogs.com/gif.latex?P=%5Cfrac%7BK%7D%7B%5CDelta%20t%7D]]

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> ||

<span style="color: #ff4000; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Short Overview of Power in an AC Circuit
<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">In an electric current, power represents the rate of energy flow passing a certain point of a current. In an alternating current circuit (AC circuit), power isn't lost as a result of pure inductors and pure capacitors, where pure inductors don't have any capacitance or resistance, and pure capacitors don't have any inductance or resistance. In an AC Circuit that contains only a generator and capacitor, a voltage drop is shown when the capacitor gathers charge, where the energy stored in the capacitor when the voltage reaches its maximum value is <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The charge leaves the capacitor plates and returns to the voltage source when the current reverses direction, where the capacitor is being charged during one-half of each cycle and the charge is being returned to the voltage source during the other half. In an AC circuit no power losses occur in a capacitor because the average power supplied by the source is zero. The energy stored in the inductor is a maximum when the current reaches its maximum value; it's represented using <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">In an AC Circuit, power is calculated using the formula <span style="font-family: 'Times New Roman',Times,serif; font-size: 150%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">, <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">where the quantity I represents the current in units of Coulomb/second, or Amperes (Amp), and the quantity V represents voltage in units of volts (V). These quantities in an AC circuit are continuously varying, so the more preferred measurement of power in an AC circuit would be the average power.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Average Power in an AC Circuit
<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">In an AC circuit, average power, the average wattage of power flowing through a current during a certain given time, is calculated using the formula <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">, <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">where //V// is the voltage, //I// is the current, and //φ// is the phase angle between the voltage and current. The power factor is the quantity //cosφ//. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Power
<span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://auto.howstuffworks.com/auto-parts/towing/towing-capacity/information/fpte7.htm] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://www.jfinternational.com/ph/power.html] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://www.physicsclassroom.com/class/energy/u5l1e.cfm] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://scienceworld.wolfram.com/physics/Power.html] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://suite101.com/article/] <span style="color: #ff4000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|energy-and-power-in-physics-a49740]

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Power in an AC Circuit
<span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://www.allaboutcircuits.com/vol_2/chpt_11/1.html] <span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[|http://hyperphysics.phy-astr.gsu.edu/hbase/electric/powerac.htm]

<span style="color: #808080; font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Sample Problems
<span style="color: #000000; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[] <span style="color: #808080; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[] <span style="color: #808080; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[] <span style="color: #808080; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[] <span style="color: #808080; font-family: 'Courier New',Courier,monospace; font-size: 110%;">[]

<span style="color: #ff4000; font-family: Arial,Helvetica,sans-serif; font-size: 150%;">References
<span style="color: #ff4000; display: block; font-family: Arial,Helvetica,sans-serif; font-size: 130%; height: 100%; position: relative; width: 100%;">//<span style="color: #ffffff; cursor: pointer; display: block; font-family: lucida grande,tahoma,verdana,arial,sans-serif; font-size: 10px; height: 16px; position: absolute; right: -20px; top: -10px; width: 16px;">X //
 * 1) "AC power". //wikipedia.com.// n.d. Web 1 June 2012. <http://en.wikipedia.org/wiki/AC_power>.
 * 2) <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> //hyperphysics.// n.d. Web. 1 June 2012. < http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html>.
 * 3) "Power (physics)". //wikipedia.com.// n.d. Web. 1 June 2012. <http://en.wikipedia.org/wiki/Power_%28physics%29>.
 * 4) Serway, Raymond A., Jerry S. Faughn, Chris Vuille, and Charles A. Benett. //College Physics.// 7th ed. Thomson Brooks/Cole, 2006. Print.

<span style="color: #ffffff; cursor: pointer; display: block; font-family: 'lucida grande',tahoma,verdana,arial,sans-serif; font-size: 10px; height: 16px; position: absolute; right: -20px; top: -10px; width: 16px;">X

<span style="color: #ffffff; cursor: pointer; display: block; font-family: lucida grande,tahoma,verdana,arial,sans-serif; font-size: 10px; height: 16px; position: absolute; right: -20px; top: -10px; width: 16px;">X

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