TOPIC 5: Work, Energy & Power

WORK Definition of Work: When a force causes a displacement of an object Components of the force need to be in the direction of the displacement

Net Work done by a Constant Net Force Work = Force (F) x Displacement (x) W = Fx W = Fx = (Fcosθ)x ** Only the component of the force in the direction of the displacement, contributes to work

Units of Work Work = Force x Displacement = Newtons x meters Newton x meter  Joule (J) * Joule is named after James Prescott Joule (1818-1889) who made major contributions to the understanding of energy, heat, and electricity =

Work Work: Scalar quantity Can be positive or negative Positive work  Exists when the force & displacement vectors point in the same direction Negative work  Exists when the force & displacement vectors point in opposite directions Ex. If you kick a stationary soccer ball, propelling it downfield, you have done positive work on the ball because the force and the displacement are in the same direction. When a goalie catches a kicked ball, negative work is done by the force from the goalie’s hands on the ball. The force on the ball is in the opposite direction of the ball’s displacement, with the result that the ball slows down.

Problem How much work is done on a vacuum cleaner pulled 3 m by a force of 50 N at an angle of 30° above the horizontal? W = (Fcosθ)x W = ? F = 50N d = 3m θ = 30° W = (50N)(cos30°)(3m) = 130 J

ENERGY Kinetic Energy: * Energy associated with an object in motion * Depends on speed and mass * Scalar quantity * SI unit for all forms of energy = Joule (J) KE = ½ mv2 KE = ½ x mass x (velocity)2 There are many different types of energy- Elastic potential energy, kinetic energy, nuclear, electrical, chemical, mechanical, etc.

Kinetic Energy If a bowling ball and a soccer ball are traveling at the same speed, which do you think has more kinetic energy? KE = ½ mv2 * Both are moving with identical speeds * Bowling ball has more mass than the soccer ball  Bowling ball has more kinetic energy Kinetic energy is proportional to the mass of the moving object. If it moved at the same speed, an arrow twice as massive as this one would have twice as much kinetic energy

Kinetic Energy Problem A 7 kg bowling ball moves at 3 m/s. How fast must a 2.45 g tennis ball move in order to have the same kinetic energy as the bowling ball? Velocity of tennis ball = 160 m/s

Work-Kinetic Energy Theorem Net work done on a particle equals the change in its kinetic energy (KE) W = ΔKE Kinetic Books 7.7 Consider the foot kicking the soccer ball in Concept 1. We want to relate the work done by the force exerted by the foot on the ball to the ball’s change in kinetic energy. To focus solely on the work done by the foot, we ignore other forces acting on the ball, such as friction. Initially, the ball is stationary. It has zero kinetic energy because it has zero speed. The foot applies a force to the ball as it moves through a short displacement. This force accelerates the ball. The ball now has a speed greater than zero, which means it has kinetic energy. The work-kinetic energy theorem states that the work done by the foot on the ball equals the change in the ball’s kinetic energy. In this example, the work is positive (the force is in the direction of the displacement) so the work increases the kinetic energy of the ball. As shown in Concept 2, a goalie catches a ball kicked directly at her. The goalie’s hands apply a force to the ball, slowing it. The force on the ball is opposite the ball’s displacement, which means the work is negative. The negative work done on the ball slows and then stops it, reducing its kinetic energy to zero. Again, the work equals the change in energy; in this case, negative work on the ball decreases its energy.

PROBLEM What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg. SEE Kinetic Books 7.7 Draw scenario on board…Give values F =240N, x =0.20m W = F ∙ Δx W = (240 N) (0.20 m) = 48 J W = ΔKE = 48 J KE = ½ mv2 = 48 J v2 = 2(48 J)/0.42 kg v = 15 m/s

PROBLEM What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg. W = F ∙ Δx W = (240 N) (0.20 m) = 48 J W = ΔKE = 48 J KE = ½ mv2 = 48 J v2 = 2(48 J)/0.42 kg v = 15 m/s SEE Kinetic Books 7.7 Draw scenario on board…Give values F =240N, x =0.20m Kinetic bks hmwk 7.9, 7.10, 7.11

Work-Kinetic Energy Theorem On a frozen pond, a person kicks a 10 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10? m = 10 kg vi = 2.2 m/s vf = 0 m/s μk = 0.10 d = ? CHALLENGING! Probably skip???

Work-Kinetic Energy Theorem Wnet = Fnetdcosθ * Net work done of the sled is provided by the force of kinetic friction Wnet = Fkdcosθ  Fk = μkN  N = mg Wnet = μkmgdcosθ * The force of kinetic friction is in the direction opposite of d  θ = 180° * Sled comes to rest  So, final KE = 0 Wnet = Δ KE = ½ mv2f – ½ mv2i Wnet = -1/2 mv2i CHALLENGING!

Work-Kinetic Energy Theorem Use the work-kinetic energy theorem, and solve for d Wnet = ΔKE - ½ mv2i = μkmgdcosθ d = 2.5 m CHALLENGING!

POWER POWER: * A quantity that measures the rate at which work is done or energy is transformed * Power = work / time interval P = W/Δt (W = Fx  P = Fx/Δt  v = x/Δt) * Power = Force x speed P = Fv

POWER SI Unit for Power: Watt (W)  Defined as 1 joule per second (J/s) Horsepower = Another unit of power 1 hp = 746 watts

POWER PROBLEM A 193 kg curtain needs to be raised 7.5 m, in as close to 5 s as possible. The power ratings for three motors are listed as 1 kW, 3.5 kW, and 5.5 kW. What motor is best for the job?

POWER PROBLEM m = 193 kg Δt = 5s d =7.5m P = ? P = W/Δt = Fx/Δt = mgx/Δt = (193kg)(9.8m/s2)(7.5m)/5s = 280 W  2.8 kW ** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the curtain fast enough, and the 5.5 kW motor will lift the curtain too fast

POTENTIAL ENERGY Potential Energy: * Stored energy * Associated with an object that has the potential to move because of its position relative to some other location Example: Balancing rock- Arches National Park, Utah Delicate Arch- Arches National Park, Utah

GRAVITATIONAL POTENTIAL ENERGY- Definition Gravitational potential energy PEg is the energy an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level: PEg = mgh SI Unit = Joule (J)

Problem What is the bucket’s gravitational potential energy? Kinetic Books 7.13- Need to supply students with details and drawing PE = mgh PE = (2.00 kg)(9.80 m/s2)(4.00 m) PE = 78.4 J

Problem What is the bucket’s gravitational potential energy? PE = mgh PE = (2.00 kg)(9.80 m/s2)(4.00 m) PE = 78.4 J PE = mgh PE = (2.00 kg)(9.80 m/s2)(4.00 m) PE = 78.4 J

Gravitational Potential Energy Example: A Gymnast on a Trampoline The gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast? CHALLENGING- Perhaps SKIP???

Gravitational Potential Energy CHALLENGING- Perhaps SKIP???

Elastic Potential Energy * Energy stored in any compressed or stretched object Spring, stretched strings of a tennis racket or guitar, rubber bands, bungee cords, trampolines, an arrow drawn into a bow, etc.

Springs When an external force compresses or stretches a spring  Elastic potential energy is stored in the spring The more stretch, the more stored energy For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); Constant of proportionality is known as the spring constant (k) Fspring = k * x

Hooke’s Law If a spring is not stretched or compressed  no potential energy is being stored Spring is in an Equilibrium position Equilibrium position: Position spring naturally assumes when there is no force applied to it Zero potential energy position

Hooke’s Law Special equation for springs Relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant PE elastic = ½kx2 k = Spring constant (N/m) Stiffer the spring  Larger the spring constant x = Amount of compression relative to the equilibrium position

Potential Energy Problem A 70 kg stuntman is attached to a bungee cord with an unstretched length of 15 m. He jumps off the bridge spanning a river from a height of 50m. When he finally stops, the cord has a stretched length of 44 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, what is the total potential energy relative to the water when the man stops falling?

Potential Energy Problem * Zero level for gravitational potential energy is chosen to be the surface of the water * Total potential energy  sum of the gravitational & elastic potential energy PEtotal = PEg + PEelastic = mgh + ½ kx2 * Substitute the values into the equation PEtotal = 3.43 x 104 J

Potential Energy The energy stored in an object due to its position relative to some zero position An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

Linking Work to Mechanical Energy WORK is a force acting upon an object to cause a displacement When work is done upon an object, that object gains energy Energy acquired by the objects upon which work is done is known as MECHANICAL ENERGY Weightlifter applies a force to cause a barbell to be displaced. The barbell then possesses mechanical energy- all in the form of potential energy

Mechanical Energy Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position Discuss Examples: Moving car possesses mechanical energy due to its motion (kinetic energy) A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitational potential energy)

Total Mechanical Energy *Total Mechanical Energy: The sum of kinetic energy & all forms of potential energy 1. Kinetic Energy (Energy of motion) KE = ½ mv2 2. Potential Energy (Stored energy of position) a. Gravitational PEg = mgh b. Elastic PEelastic = ½ kx2 The sum of kinetic energy & all the forms of potential energy (gravitational and elastic) Skiing problem

Mechanical Energy CONSERVATION OF MECHANICAL ENERGY: * In the absence of friction, mechanical energy is conserved, so the amount of mechanical energy remains constant MEi = MEf Initial mechanical energy = final mechanical energy (in the absence of friction) PEi + KEi = PEf + KEf mghi + ½ mvi2 = mghf + ½ mvf2

Conservation of Energy Problem Starting from rest, a child zooms down a frictionless slide from an initial height of 3 m. What is her speed at the bottom of the slide? (Assume she has a mass of 25 kg)

Conservation of Energy Problem hi = 3m m = 25kg vi = 0 m/s hf = 0m vf = ? Slide is frictionless  Mechanical energy is conserved Kinetic energy & potential energy = only forms of energy present KE = ½ mv2 PEg = mgh Final gravitational potential energy = zero (Bottom of the slide)  PEgf = 0 Initial gravitational potential energy  Top of the slide  PEgi = mghi  (25kg)(9.8m/s2)(3m) = 736 J

Conservation of Energy Problem hi = 3m m = 25kg vi = 0 m/s hf = 0m vf = ? Initial Kinetic Energy = 0, because child starts at rest KEi = 0 Final Kinetic Energy KEf = ½ mv2  ½ (25kg)v2f MEi = MEf PEi + KEi = PEf + Kef 736 J + 0 J = 0 J + (1/2)(25kg)(v2f) vf = 7.67 m/s

Mechanical Energy  Ability to do Work An object that possesses mechanical energy is able to do work Its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced Classic Example  Massive wrecking ball of a demolition machine

I did not use anymore of these slides 2011

Mechanical Energy is the ability to do work… An object that possesses mechanical energy (whether it be kinetic energy or potential energy) has the ability to do work That is… its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced

Mechanical Energy Work is a force acting on an object to cause a displacement In the process of doing work  the object which is doing the work exchanges energy with the object upon which the work is done When work is done up the object  that object gains energy

Mechanical Energy A weightlifter applies a force to cause a barbell to be displaced Barbell now possesses mechanical energy- all in the form of potential energy ** The energy acquired by the objects upon which work is done is known as mechanical energy

Mechanical Energy is the ability to do work… Examples on website: Massive wrecking ball of a demolition machine The wrecking ball is a massive object which is swung backwards to a high position and allowed to swing forward into a building structure or other object in order to demolish it Upon hitting the structure, the wrecking ball applies a force to it in order to cause the wall of the structure to be displaced Mechanical energy = ability to do work

Work- Energy Theorem Categorize forces based upon whether or not their presence is capable of changing an object’s total mechanical energy * Certain types of forces, which when present and when involved in doing work on objects, will change the total mechanical energy of the object * Other types of forces can never change the total mechanical energy of an object, but rather only transform the energy of an object from PE to KE or vice versa ** Two categories of forces  Internal & External

Work- Energy Theorem External Forces: Applied force, normal force, tension force, friction force and air resistance force Internal Forces: Gravity forces, spring forces, electrical forces and magnetic forces

Work- Energy Theorem THE BIG CONCEPT!! * When the only type of force doing net work upon an object is an internal force (gravitational and spring forces)  Total mechanical energy (KE + PE) of that object remains constant  Object’s energy simply changes form  Conservation of Energy ** Ex) As an object is “forced” from a high elevation to a lower elevation by gravity  Some of the PE is transformed into KE (Yet, the sum of KE + PE = remains constant)

Work- Energy Theorem THE BIG CONCEPT!! * If only internal forces are doing work  energy changes forms (KE to PE or vice versa)  total mechanical energy is therefore conserved * Internal forces – referred to as conservative forces Quick Quiz

Work-Energy Relationship Analysis of situations in which work is conserved  only internal forces are involved TMEi + WEXT = TMEf (Initial amount of total mechanical energy (TMEi) plus the work done by external forces (WEXT)  equals the final amount of total mechanical energy (TMEf)) KEi + PEi + Wext = KEf + PEf KEi + PEi = KEf + Pef Website

Work- Energy Theorem THE BIG CONCEPT!! * Forces are categorized as being either internal or external based upon the ability of that type of force to change an object’s total mechanical energy when it does work upon an object * Net work done upon an object by an external force  Changes the total mechanical energy (KE + PE) of the object  Positive work = object gained energy  Negative work = object lost energy

Work- Energy Theorem THE BIG CONCEPT!! * Gain or loss in energy can be in the form of  PE, KE, or both Under such circumstances, the work which is done is equal to the change in mechanical energy of the object ** External forces  capable of changing the total mechanical energy of an object (Nonconservative forces)

Work-Energy Relationship Analysis of situations involving external forces TMEi + WEXT = TMEf (Initial amount of total mechanical energy (TMEi) plus the work done by external forces (WEXT)  equals the final amount of total mechanical energy (TMEf)) KEi + PEi + Wext = KEf + PEf Practice Problems

DEFINITION OF A CONSERVATIVE FORCE Version 1 A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions. Version 2 A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point.

Conservative Versus Nonconservative Forces

Conservative Versus Nonconservative Forces Version 1 A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions. Work done is due to the change in gravitational potential energy

Conservative Versus Nonconservative Forces Version 2 A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point.

Conservative Versus Nonconservative Forces An example of a nonconservative force is the kinetic frictional force. The work done by the kinetic frictional force is always negative. Thus, it is impossible for the work it does on an object that moves around a closed path to be zero. The concept of potential energy is not defined for a nonconservative force.

Conservative Versus Nonconservative Forces In normal situations both conservative and nonconservative forces act simultaneously on an object, so the work done by the net external force can be written as

Conservative Versus Nonconservative Forces THE WORK-ENERGY THEOREM

The Conservation of Mechanical Energy If the net work on an object by nonconservative forces is zero, then its energy does not change:

The Conservation of Mechanical Energy THE PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY The total mechanical energy (E = KE + PE) of an object remains constant as the object moves, provided that the net work done by external nonconservative forces is zero.

The Conservation of Mechanical Energy

The Conservation of Mechanical Energy Example A Daredevil Motorcyclist A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff at 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

The Conservation of Mechanical Energy

The Conservation of Mechanical Energy

Nonconservative Forces and the Work-Energy Theorem

Nonconservative Forces and the Work-Energy Theorem Example Fireworks Assuming that the nonconservative force generated by the burning propellant does 425 J of work, what is the final speed of the rocket. Ignore air resistance. The mass of the rocket is 0.2kg.

Nonconservative Forces and the Work-Energy Theorem

POWER POWER: * A quantity that measures the rate at which work is done or energy is transformed * Power = work / time interval P = W/Δt W = Fd P = Fd/Δt  v = d/Δt * Power = Force x speed P = Fv

POWER SI Unit for Power: Watt (W)  Defined as 1 joule per second (J/s) Horsepower = Another unit of power 1 hp = 746 watts

POWER PROBLEM A 193 kg curtain needs to be raised 7.5 m, in as close to 5 s as possible. The power ratings for three motors are listed as 1 kW, 3.5 kW, and 5.5 kW. What motor is best for the job?

POWER PROBLEM m = 193 kg Δt = 5s d =7.5m P = ? P = W/Δt = Fd/Δt = mgd/Δt = (193kg)(9.8m/s2)(7.5m)/5s = 280 W  2.8 kW ** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the curtain fast enough, and the 5.5 kW motor will lift the curtain too fast

THE PRINCIPLE OF CONSERVATION OF ENERGY Energy can neither be created nor destroyed, but can only be converted from one form to another. * Disclaimer: This powerpoint presentation is a compilation of various works.

Question A cart is loaded with a brick and pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is 3.0 kg and the height of the seat top is 0.45 meters, then what is the potential energy of the loaded cart at the height of the seat-top? PE = m*g*h PE = (3 kg ) * (9.8 m/s/s) * (0.45m) PE = 13.2 J

Question If a force of 14.7 N is used to drag the loaded cart (from previous question) along the incline for a distance of 0.90 meters, then how much work is done on the loaded cart? W = F * d * cos Theta W = 14.7 N * 0.9 m * cos (0 degrees) W = 13.2 J (Note: The angle between F and d is 0 degrees because the F and d are in the same direction)