Work… In everyday speech work has a very general meaning. In describing motion in physics, work has a very specific meaning. In everyday speech work has a very general meaning. In describing motion in physics, work has a very specific meaning.

Chair Example StandingWalking No work is done on the chair

Work is defined as the product of the force applied to cause motion and the distance the object moves in the direction of the force. Work is defined as the product of the force applied to cause motion and the distance the object moves in the direction of the force. Work is done only when components of a force are parallel to a displacement Work is done only when components of a force are parallel to a displacement

FORMULA W = fd IN DIRECTION OF MOTION The symbol for work is W The symbol for work is W Work has 2 acceptable units Work has 2 acceptable units Nm Nm Joules (J) Joules (J)

Lifting an apple about 2ft is a Joule Lifting an apple about 2ft is a Joule 3 good push-ups is about 1000J 3 good push-ups is about 1000J JOULE

WORK In order for work to be done, three things are necessary: There must be an applied force. The force must act through a certain distance, called the displacement. The force must have a component along the displacement.

Work is a scalar quantity equal to the product of the magnitudes of the displacement and the component of the force in the direction of the displacement. W = F. x or W = F cos  x UNITS: N.m this unit is called a Joule (J)

As long as this person does not lift or lower the bag of groceries, he is doing no work on it. The force he exerts has no component in the direction of motion. Work done by forces that oppose the direction of motion, such as friction, will be negative.

Centripetal forces do no work, as they are always perpendicular to the direction of motion.

If the force acting on an object varies in magnitude and/or direction during the object’s displacement, graphical analysis can be used to determine the work done. The Force is plotted on the y-axis and the distance through which the object moves is plotted on the x-axis. The work done is represented by the area under the curve.

5.1 A push of 200 N moves a 100 N block up a 30  inclined plane. The coefficient of kinetic friction is 0.25 and the length of the plane is 12 m. a. Find the work done by each force acting on the block. F A = 200 N F G = 100 N θ = 30˚ μ = 0.25 x = 12 m Forces acting: F f F A F G and F N F N does NO work. W

FNFN FfFf FGFG F Gy F Gx θ W FA = F A x = 200 (12) = 2400 J F f = μ F N = μ F G cos 30˚ = 0.25 (100) cos 30˚ = 21.6 N W Ff = - F f x = (12) = J W FG = F G x = -F Gx x = - F G sin30˚x = sin 30˚ (12) = J FAFA

FNFN FfFf FGFG F Gy F Gx θ b. Show that the net work done by these forces is the same as the work of the resultant force. Net work: ΣW = = J The resultant force: ΣF x = F A - F f - F Gx = = N W F = F x. x = (12) = J FAFA

Positive Work Negative Work No Work Force is in the direction of motion Force opposes motion Force is 90° to motion Object is not in motion Situations that affect the sign of work

10N Moves 2m W = fd W = 20J of Work Notice direction of motion is the same as the applied force

60° 10N X Y 2m How would you solve this? Force applied is NOT in the same direction as the objects motion. Think back to vectors and use the component of the force applied in the direction the object moves.

60° 10N X Y 2m COS θ = adj/ hyp COS 60° = force parallel to motion 10N 10N force para. = COS 60° (10N) force para. = 5N w = F(parallel) D 5N (2m) = 10Nm

W = Fd (COS θ) Always measure angle with horizontal! The above formula works in every case θ = 0° θ = 90° No work because no motion in direction of force

Work The VERTICAL component of the force DOES NOT cause the block to move the right. The energy imparted to the box is evident by its motion to the right. Therefore ONLY the HORIZONTAL COMPONENT of the force actually creates energy or WORK. When the FORCE and DISPLACEMENT are in the SAME DIRECTION you get a POSITIVE WORK VALUE. The ANGLE between the force and displacement is ZERO degrees. What happens when you put this in for the COSINE? When the FORCE and DISPLACEMENT are in the OPPOSITE direction, yet still on the same axis, you get a NEGATIVE WORK VALUE. This negative doesn't mean the direction!!!! IT simply means that the force and displacement oppose each other. The ANGLE between the force and displacement in this case is 180 degrees. What happens when you put this in for the COSINE? When the FORCE and DISPLACEMENT are PERPENDICULAR, you get NO WORK!!! The ANGLE between the force and displacement in this case is 90 degrees. What happens when you put this in for the COSINE?

The Work Energy Theorem Up to this point we have learned Kinematics and Newton's Laws. Let 's see what happens when we apply BOTH to our new formula for WORK! 1.We will start by applying Newton's second law! 2.Using Kinematic #3! 3.An interesting term appears called KINETIC ENERGY or the ENERGY OF MOTION!

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 in speed and thus a change in KINETIC ENERGY. Since both WORK and KINETIC ENERGY are expressed in JOULES, they are EQUIVALENT TERMS! " The net WORK done on an object is equal to the change in kinetic energy of the object."

Example W=Fxcos  A 70 kg base-runner begins to slide into second base when moving at a speed of 4.0 m/s. The coefficient of kinetic friction between his clothes and the earth is He slides so that his speed is zero just as he reaches the base (a) How much energy is lost due to friction acting on the runner? (b) How far does he slide? = N -560 J 1.17 m

Energy The Stuff that makes things move The ability to do work The ability to do work Has the units of Joules (J) Has the units of Joules (J) There are 2 kinds of mechanical energy There are 2 kinds of mechanical energy

Kinetic Energy This is the energy associated with an objects motion. This is the energy associated with an objects motion. KE depends on mass and velocity KE depends on mass and velocity When the object is treated as a particle, the formula for KE is… When the object is treated as a particle, the formula for KE is… KE = ½ mV 2 manipulated V = 2KE/m M = 2KE/V 2

KE is a scalar quantity KE is a scalar quantity The SI unit for KE is the Joule, yes the same as for work The SI unit for KE is the Joule, yes the same as for work Look at sample prob. 5B Look at sample prob. 5B Page 173 DO practice problems DO practice problems 5B 1-5 on page 174 5B 1-5 on page 174 KE is a scalar quantity KE is a scalar quantity The SI unit for KE is the Joule, yes the same as for work The SI unit for KE is the Joule, yes the same as for work Look at sample prob. 5B Look at sample prob. 5B Page 173 DO practice problems DO practice problems 5B 1-5 on page 174 5B 1-5 on page 174

Work- Kinetic Energy Theorem The net work done on an object is equal to the change in the kinetic energy of the object The net work done on an object is equal to the change in the kinetic energy of the object W net = ΔKE W net = ΔKE W net = KE final – KE initial W net = KE final – KE initial fd(cos θ) = ½ mV 2 fd(cos θ) = ½ mV 2 The KE of an object is equal to the work that moving object can do

This theorem allows us to think of KE as the work an object can do as it comes to rest, or the amount of energy contained in the moving object This theorem allows us to think of KE as the work an object can do as it comes to rest, or the amount of energy contained in the moving object The KE of the moving hammer can do work KE = Work done (net) fd = ½ mv 2 some of the energy is sound, heat and light (if spark)

Potential Energy This is the energy associated with an object due to the position of the object. This is the energy associated with an object due to the position of the object. STORED ENERGY STORED ENERGY There are two kinds of potential energy There are two kinds of potential energy 1. GRAVITATIONAL POTENTIAL ENERGY 2. ELASTIC POTENTIAL ENERGY

Gravitational Potential Energy (PE g ) The energy associated with an object due to the objects position relative to a gravitational reference The energy associated with an object due to the objects position relative to a gravitational reference Wh = PE g = mgh = mass x gravity x height acceleration Has the unit of joules gm = w

Elastic Potential Energy (PE elastic ) The energy associated with a stretched or compressed elastic object The energy associated with a stretched or compressed elastic object Spring, bungee cord, rubber band Spring, bungee cord, rubber band

Elastic Potential Energy

Overhead (springs) In both the compressed and stretched example, energy is stored In both the compressed and stretched example, energy is stored PE elastic = ½ KX 2 PE elastic = ½ KX 2 K = spring constant K = spring constant X = distance stretched or compressed X = distance stretched or compressed Practice Problems 5D 1-3 pg. 180

Energy is transferred from one form to another Energy is transferred from one form to another Pendulum PE = max KE = min PE = min KE = max PE = max KE = min As the pendulum swings, PE is transferred to KE. As the bob swings upwards KE is stored as PE

10M PE = 10 J KE = 0 J PE = 0 J KE = 10 J PE = 5 J KE = 5 J PE = mgh A falling egg Mass =.1kg Height = 10m

ENERGY Energy is that which can be converted into work. When something has energy, it is able to perform work or, in a general sense, to change some aspect of the physical world.

In mechanics we are concerned with two kinds of energy: KINETIC ENERGY: K, energy possessed by a body by virtue of its motion. Units: Joules (J) POTENTIAL ENERGY: PE, energy possessed by a system by virtue of position or condition. PE = m g h Units: Joules (J)

WORK-ENERGY PRINCIPLE: The work of a resultant external force on a body is equal to the change in kinetic energy of the body. W =  KUnits: Joules (J)

W =  PE

5.2 What average force F is necessary to stop a 16 g bullet traveling at 260 m/s as it penetrates into wood at a distance of 12 cm? v f = 0 m/s m = kg v o = 260 m/s x = 0.12 m W = ΔK = N WE

CONSERVATIVE AND NON-CONSERVATIVE FORCES The work done by a conservative force depends only on the initial and final position of the object acted upon. An example of a conservative force is gravity. The work done equals the change in potential energy and depends only on the initial and final positions above the ground and NOT on the path taken.

Friction is a non-conservative force and the work done in moving an object against a non-conservative force depends on the path. For example, the work done in sliding a box of books against friction from one end of a room to the other depends on the path taken.

For mechanical systems involving conservative forces, the total mechanical energy equals the sum of the kinetic and potential energies of the objects that make up the system and is always conserved.

A roller-coaster car moving without friction illustrates the conservation of mechanical energy.

In real life applications, some of the mechanical energy is lost due to friction. The work due to non- conservative forces is given by: W NC = ΔPE + ΔK or W NC = E f - E o

5.3 A ballistic pendulum apparatus has a 40-g ball that is caught by a 500-g suspended mass. After impact, the two masses rise a vertical distance of 45 mm. Find the velocity of the combined masses just after impact. m 1 = 0.04 kg m 2 = kg h = m K 0 = PE f COE

PE f = (m 1 +m 2 ) gh f = ( )(9.8)(0.045) = 0.24 J K 0 = PE f = 0.24 J K 0 = PE f m 1 = 0.04 kg m 2 = kg h = m = 0.94 m/s

5.4 The tallest and fastest roller coaster in the world is the Steel Dragon in Japan. The ride includes a vertical drop of 93.5 m. The coaster has a speed of 3 m/s at the top of the drop. a. Neglect friction and find the speed of the riders at the bottom.? v A = 3 m/s h A = 93.5 m h B = 0 m At point A: PE A + K A At point B: K B PE A + K A = K B A B = 42.9 m/s (about 96 mi/h) COE

b. Find the work done by non-conservative forces on a 55 kg rider during the descent if the actual velocity at the bottom is 41 m/s. W NC = E f - E 0 = K B - (PE A + K A ) v A = 3 m/s v B = 41 m/s h A = 93.5 m h B = 0 m m = 55 kg = J

5.5 A 20-kg sled rests at the top of a 30˚ slope 80 m in length. If μ k = 0.2, what is the velocity at the bottom of the incline? m = 20 kg θ = 30° r = 80 m μ k = 0.2 W NC = E f - E o = K f - PE 0 COE

m = 20 kg θ = 30° x = 80 m μ k = 0.2 x h h = x sin θ h = 80 sin 30° = 40 m PE 0 = mgh 0 = 20(9.8)(40) = 7840 J F f = μ k F N = μ k F gy = μ k F g cos30° = (0.2)(20)(9.8)cos30° = 34 N W NC = - F f r = - 34 (80) = J W NC = K f - PE 0 K f = PE 0 + W NC = = 5120 J

= 22.6 m/s

Mechanical Energy The sum of Kinetic Energy and ALL forms of Potential energy associated with an object or group of objects The sum of Kinetic Energy and ALL forms of Potential energy associated with an object or group of objects ME is not a unique form of energy. Its merely a way of classifying energy ME is not a unique form of energy. Its merely a way of classifying energy ME includes KE and PE ME includes KE and PE

Mechanical Energy ME is different from non mechanical energy (nuclear, chemical, thermal, internal, electrical) ME is different from non mechanical energy (nuclear, chemical, thermal, internal, electrical) ME = Σ KE + Σ PE ME = Σ KE + Σ PE ME = ½ mv 2 + mgh (if PE is NOT present, elastic) ME = ½ mv 2 + mgh (if PE is NOT present, elastic) Σ SPN

Conservation Of Mechanical Energy Conservation of Mechanical Energy can also be written as… Conservation of Mechanical Energy can also be written as… ME i = ME f ME i = ME f ½ mv i 2 + mgh i = ½ mv f 2 + mgh f ½ mv i 2 + mgh i = ½ mv f 2 + mgh f True when friction can be ignored True when friction can be ignored

The Law of Conservation of Energy: The total energy of a closed system is constant.

Often is the case that KE i or KE f or PE i or PE f will be zero. When that is the case… Often is the case that KE i or KE f or PE i or PE f will be zero. When that is the case… mgh = ½ mv 2 2mgh = v 2 m V = 2gh h = V 2 2g 2g

Power This quantity also has a very specific meaning in science that can be confused by common English usage This quantity also has a very specific meaning in science that can be confused by common English usage Power is the rate of doing work Power is the rate of doing work That is to say that power is the rate at which energy is transferred That is to say that power is the rate at which energy is transferred

Power Power is work done divided by the time taken to do the work Power is work done divided by the time taken to do the work Power = Work = fd P = w Time t t Time t t Power is measured in watts (W) J/s Power is measured in watts (W) J/s A watt is a small unit, 1 watt is about what is needed to lift a 2N glass of water.5m to your mouth in 1 second. A watt is a small unit, 1 watt is about what is needed to lift a 2N glass of water.5m to your mouth in 1 second.

Watts Since watts are so small, we sometimes use Kilowatts Since watts are so small, we sometimes use Kilowatts 1 KW = 1000W 1 KW = 1000W Watts are metric Watts are metric Horse power is traditional Horse power is traditional 1 Horse power = 746 Watts 1 Horse power = 746 Watts

Watts Watts are named after James Watt, the inventor of the steam engine Watts are named after James Watt, the inventor of the steam engine

Practice Problem An electric motor lifts an elevator that weighs 12000N a distance of 9m in 15sec An electric motor lifts an elevator that weighs 12000N a distance of 9m in 15sec What is the motors power in watts? What is the motors power in watts? What is the motors power in kilowatts? What is the motors power in kilowatts? Given f = 12000N d = 9m t = 15s P = ? Formula P = fd/t Solution P = 12000(9) 15 A.P = 7200 W B.7.2 KW

POWER Is the rate at which work is performed.P = work/time The difference between walking and running up these stairs is power. The change in gravitational potential energy is the same. UNITS: = Watt

5.6 A 1100-kg car starting from rest, accelerates for 5.0 s. The magnitude of the acceleration is 4.6 m/s 2. What power must the motor produce to cause this acceleration? m = 1100 kg v o = 0 m/s t = 5 s a = 4.6 m/s 2 F = ma = (1100)(4.6) = 5060 N v f = v o + at = (5) = 23 m/sThe average velocity is: 23/2 = 11.5 m/s P = Fv = (5060)(11.5) = 5.82x10 4 W

ELASTIC FORCE The force F s applied to a spring to stretch it or to compress it an amount x is directly proportional to x. F s = - k x Units: Newtons (N) Where k is a constant called the spring constant and is a measure of the stiffness of the particular spring. The spring itself exerts a force in the opposite direction:

This force is sometimes called restoring force because the spring exerts its force in the direction opposite to the displacement. This equation is known as the spring equation or Hooke’s Law.

The elastic potential energy is given by: PE s = ½ kx 2 Units: Joules (J)

5.7 A dart of mass kg is pressed against the spring of a toy dart gun. The spring (k = 250 N/m) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its normal length, what speed does the dart acquire? m = 0.1 kg k = 250 N/m x = 0.06 m PE s = K ½ kx 2 = ½ mv 2 = 3 m/s

page 108 Problem 4