Work, Power & Energy

A body builder takes 1 second to pull the strap to the position shown. If the body builder repeats the same motion in 0.5 s, does he do more work? A body builder takes 1 second to pull the strap to the position shown. If the body builder repeats the same motion in 0.5 s, does he do more work? ????????????????????????????????????????

Work Provides a link between force and energy Provides a link between force and energy The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement Scalar quantity (magnitude only) Scalar quantity (magnitude only)

Work, cont. F cos θ is the component of the force in the direction of the displacement F cos θ is the component of the force in the direction of the displacement Δ x is the displacement Δ x is the displacement If the Force is in the direction of the displacement, then cos 0 o = 1 If the Force is in the direction of the displacement, then cos 0 o = 1 Units: Newton meter = Joule N m = J

Work Examples A man pulls on a cart with a 100N force on at an angle of 25 o from the horizontal, moving it 7.5 m. How much work does he do? A forklift lifts a 2000 N crate at a constant speed. If the total work done on the crate is 7000 J, what height is the crate lifted? = (100N) (7.5 m) cos 25 o = 680 N m or J 7000 J= (2000N) s cos 0 o S = 3.5 m

More About Work, cont. Work can be positive or negative Work can be positive or negative Positive if the force and the displacement are in the same direction (pushing or pulling an object) Positive if the force and the displacement are in the same direction (pushing or pulling an object) Negative if the force and the displacement are in the opposite direction (Friction slowing an object down, a ball rising in the air being slowed by gravity) Negative if the force and the displacement are in the opposite direction (Friction slowing an object down, a ball rising in the air being slowed by gravity)

When Work is Zero If the Displacement of an object is horizontal and a vertical force is applied to keep it a constant height above the ground If the Displacement of an object is horizontal and a vertical force is applied to keep it a constant height above the ground No vertical work is done, only horizontal work No vertical work is done, only horizontal work If a force is applied and the object does not move, no work is done If a force is applied and the object does not move, no work is done

Work Done by Varying Forces The work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of Force versus distance The work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of Force versus distance

Power Often also interested in the rate at which the energy transfer takes place Often also interested in the rate at which the energy transfer takes place Power is defined as this rate of work done or energy transfer Power is defined as this rate of work done or energy transfer Unit is J /s or Watt (W) Unit is J /s or Watt (W) FYI: 1 HP = 746 W FYI: 1 HP = 746 W

Power and graphs Slope of a Work vs Time Graph equals the Power developed Slope of a Work vs Time Graph equals the Power developed Slope = Power

Power Examples A machine applies a force of 1000 N over a distance of 10m in 4 seconds. What is the power rating of this machine? How long does it take a motor with a power rating of 1500 W to do 375 J of Work? = 2500 N m /s or J/s or W t = 0.25 s

A body builder takes 1 second to pull the strap to the position shown. If the body builder repeats the same motion in 0.5 s, does he do more work? A body builder takes 1 second to pull the strap to the position shown. If the body builder repeats the same motion in 0.5 s, does he do more work? ???????????????????????????????????????? Same amount of work done in both cases, but more powerful in the 2 nd case

Who is most powerful? All of our contestants will clime a flight of stairs that are 4m in height. All of our contestants will clime a flight of stairs that are 4m in height. ContestantWeightTimePower

A.The angle between the string and the tangential direction of motion at any instant is 90°.  There is no component of the tension acting in the direction of the motion.  Zero work is done by the tension on the ball. A ball is swung on a string in a horizontal circular motion. How much work is done on the ball by the tension in the string? Explain your answer.

Potential Energy Potential energy is associated with the position of the object within some system Potential energy is associated with the position of the object within some system Potential energy is a property of the system, not the object Potential energy is a property of the system, not the object A system is a collection of objects or particles interacting via forces or processes that are internal to the system A system is a collection of objects or particles interacting via forces or processes that are internal to the system

Gravitational Potential Energy Gravitational Potential Energy (  E p ) is the energy associated with the relative position of an object in space near the Earth’s surface Gravitational Potential Energy (  E p ) is the energy associated with the relative position of an object in space near the Earth’s surface Objects interact with the earth through the gravitational force Objects interact with the earth through the gravitational force Actually the potential energy of the earth- object system Actually the potential energy of the earth- object system M = Mass (kg) g = acceleration due to gravity (9.81m/s 2 on Earth)  h = change in height (m)

Reference Levels for Gravitational Potential Energy A location where the gravitational potential energy is zero must be chosen for each problem A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the potential energy is the important quantity The choice is arbitrary since the change in the potential energy is the important quantity Choose a convenient location for the zero reference height Choose a convenient location for the zero reference height often the Earth’s surface often the Earth’s surface may be some other point suggested by the problem may be some other point suggested by the problem

PRACTICE: Consider a crane which lifts a 2000-kg weight 18 m above its original resting place. What is the change in gravitational potential energy of the weight? SOLUTION:  The change in gravitational potential energy is just ∆E P = mg∆h = 2000(9.8)(18) = J. FYI  Note that the unit for ∆E P are those of work Gravitational potential energy ∆E P = mg∆h gravitational potential energy change

Hooke’s Law: Force on a spring F = - k x F = - k x F is the restoring force, which is the force needed to return the spring to its equilibrium or rest position (x) F is the restoring force, which is the force needed to return the spring to its equilibrium or rest position (x) –F is in the opposite direction of x, hence the negative sign ( – sign is ignored in most calculations) k is the spring constant k is the spring constant –K depends on how the spring was formed, the material it is made from, thickness of the wire, etc. –The greater the spring constant, the tougher the spring is to stretch/compress

Sketching and interpreting force – distance graphs EXAMPLE: A force vs. displacement plot for a spring is shown. Find the value of the spring constant, and find the spring force if the displacement is 85 cm. SOLUTION:  The slope of the graph is k  F /  x = (20N – 0 )/ (0.4m - 0)  k = 50 N m -1.  F = kx = (50N/m)(0.85m)=42.5N. F / N x/cm

Work done using a force – distance graph EXAMPLE: A force vs. displacement plot for a spring is shown. Find the work done by you if you displace the spring from 0 to 40 mm. SOLUTION:  The area under the F vs. x graph represents the work done by that force.  The area desired is from 0 mm to 40 mm, shown here: A = (1/2)bh = (1/2)(40  m)(20 N) = 0.4 J. F / N x/mm

Potential Energy in a Spring Elastic Potential Energy Elastic Potential Energy Energy stored in a spring to compress or stretch a spring from its equilibrium (rest) position to some final position x Energy stored in a spring to compress or stretch a spring from its equilibrium (rest) position to some final position x E p = Potential Energy (J) k = spring constant (N/m)  x= change in length (m)

Potential Energy of a Spring Examples The elastic force constant of a spring in a toy is 550 N/m. If the spring is compressed 10 cm, compute the potential energy stored in the spring. = ½ (550 N/m) (0.1m) 2 = 2.75 N A total of 56 J of potential energy is stored in a spring that is stretched 0.5 m. Determine the spring constant of the spring. 56 J = ½ k (0.5m) 2 k= 448 N/m

 Kinetic energy E K is the energy of motion.  The bigger the speed v, the bigger E K.  The bigger the mass m, the bigger E K.  The formula for E K, justified later E k = ½ mv 2  Looking at the units for E K we have kg(m/s) 2 = kg m 2 s -2 = (kg m s -2 )m.  In the parentheses we have a mass times an acceleration which is a Newton.  Thus E K is measured in (N m), which are (J).  Many books use KE instead of E K for kinetic energy. Kinetic Energy

Kinetic Energy Examples PRACTICE #1: What is the kinetic energy of a 40 gram bullet traveling at 400 m/s? SOLUTION:  Convert grams to kg to get m = 0.04 kg.  Then E K = ½ mv 2 =(1/2)(.04kg)(400m/s) 2 = 3200 J.

Practice #2 The kinetic energy of a 100 kg soldier is 1800J. How fast is he running at 6 m/s? Kinetic Energy Examples SOLUTION:  Then E K = ½ mv J = ½ (100kg)v 2 v= 6 m/s

Kinetic energy and momentum are related in the following way:  Since E k = ½ mv 2  Multiplying both sides by m/m  (m/m) E k = (m/m) ½ mv 2  E k = ½ m 2 v 2 /m  E k =1/2 p 2 /m = p 2 / 2m Kinetic Energy and Momentum

Momentum and Kinetic Energy Example An object with a mass of 10kg has a Potential energy of 500J. What is the magnitude of the object’s momentum? 500J = p 2 / 2 (10kg) = p kg m/s = p

 It is no coincidence that work and kinetic energy have the same units. Observe the following derivation. v 2 = u 2 + 2as (½ m)v 2 = (½ m)(u 2 + 2as) ½ mv 2 = ½ mu 2 + mas ½ mv 2 = ½ mu 2 + Fs E K final = E K initial + Work E K final - E K initial = Work ∆E K = W Work done as energy transfer FYI  This is called the Work- Kinetic Energy theorem.  It is not in the Physics Data Booklet, and I would recommend that you memorize it!

PRACTICE: Consider a crane which lifts a 2000-kg weight 18 m above its original resting place. How much work does the crane do? SOLUTION:  The force F = mg = 2000(9.8) = N .  The displacement s = 18 m .  Then W = Fs = 19620N (18m) = J.  Note that the work done by the crane is equal to the change in potential energy.. Revisiting the crane example…

EXAMPLE: Consider a crane which lifts a 2000-kg weight 18 m above its original resting place. If the cable breaks at the top, find the speed and kinetic energy of the mass at the instant it reaches the ground. SOLUTION: a = g because it is free falling. v 2 = u 2 + 2as v 2 = (-9.81m/s 2 )(18m) = v = m s -1. E K = (1/2)mv 2 = ½ (2000kg)(18.79m/s 2 ) = 353,064 J. Revisiting the crane example (part 2)  Note that the Kinetic Energy of the weight is about equal to the  E p and the work done by the crane

Elastic Potential Energy Revisited EXAMPLE: Show that the energy “stored” in a stretched or compressed spring is given by the E P = (1/2)k  x 2 SOLUTION:  As we learned, the area under the F vs. x graph gives the work done by the force during that displacement.  From F = kx and from A = (1/2) bh we obtain E P = Work = Area = (1/2)xF = (1/2)x(kx) = (1/2)kx 2. F x

The Work Energy Theorem And so what we really have is called the WORK-ENERGY THEOREM. It basically means that if we impart work to an object it will undergo a CHANGE ENERGY. Since both WORK and ENERGY are expressed in JOULES, they are EQUIVALENT TERMS! The WORK done on or by an object is equal to the change in kinetic or potential energy of the object. Work =  E p =  E k

Conservation of Mechanical Energy Conservation in general Conservation in general To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant In an isolated or ideal system, the total mechanical energy (E k + E p ) remains constant (is conserved.) In an isolated or ideal system, the total mechanical energy (E k + E p ) remains constant (is conserved.) Ideal systems: Total ME before = Total ME after Ideal systems: Total ME before = Total ME after Energy is often lost to internal or thermal energy due to friction Energy is often lost to internal or thermal energy due to friction

Example Suppose the woman in the figure above applies a 50 N force to a 25-kg box at an angle of 30 degrees above the horizontal. She manages to pull the box 5 meters. a)Calculate the WORK done by the woman on the box b)The speed of the box after 5 meters if the box started from rest J 4.16 m/s

ENERGY IS CONSERVED The law of conservation of mechanical energy states: Energy cannot be created or destroyed, only transformed! Energy BeforeEnergy After Am I moving? If yes, K o Am I above the ground? If yes, U o Am I moving? If yes, K Am I above the ground? If yes, U

Conservation of Energy A In Figure A, a pendulum is released from rest at some height above the ground position. It has only potential energy. B In Figure B, a pendulum is still above the ground position, yet it is also moving. It has BOTH potential energy and kinetic energy. C In Figure C, a pendulum is at the ground position and moving with a maximum velocity. It has only kinetic energy. D In Figure D, the pendulum has reached the same height above the ground position as A. It has only potential energy.

Energy consistently changes forms

PositionmvUKME 160 kg8 m/s Am I above the ground? Am I moving? NO, h = 0, U = 0 J 0 J Yes, v = 8 m/s, m = 60 kg 1920 J (= U+K) 1920 J

Energy consistently changes forms PositionmvUKME 160 kg8 m/s0 J1920 J 260 kg Energy Before = Energy After KOKO = U + K 1920= (60)(9.8)(1) + (.5)(60)v = v J 1332 = 30v = v 2 v = 6.66 m/s 6.66 m/s 1920 J 1332 J

Energy consistently changes forms PositionmvUKME 160 kg8 m/s0 J1920 J 260 kg6.66 m/s588 J1332 J1920 J 360 kg1920 J Am I moving at the top?No, v = 0 m/s 0 m/s0 J1920 J E B = E A Using position 1 K o = U 1920 = mgh 1920 =(60)(9.8)h h = 3.27 m

Efficiency Efficiency is defined as the ratio of the useful output to the total input This can be calculated using work, energy or power values as long as you are consistent Energy is often lost to friction (internal energy)

Sankey diagrams Incandescent bulb Efficiency: Use arrows to represent forms of energy. Width of arrows represents the amount of energy.

Sankey Diagrams LED Efficiency: