Energy and Transformation

• chemical fuel energy vehicle motion

• electric energy turning mixer, drill, etc.

• wind turbine electrical energy turn mixer

Energy: The work that a physical system is capable of doing in changing from its actual state to a specified reference state … (American Heritage Dictionary)

Energy: The capacity to do work. (Physics)

What is Work?

Some Definitions

Work

• Work is force x distance.

• It takes energy to do work.

• Less stored energy is available after productive work is done.

Physics Definition of Work

Work, W SI Unit: J = (N)(m)

Work is the useful part of a force times the distance the object moves (“s”)

“useful” means in direction of motion

sFW )cos(

Example of Work

Work = Fcosx = (80N)(cos40)(11m) = 674 J

Given: F = 80N, Angle is 40°, x is 11m,

Energy

• Kinetic, K: energy of motionK = ½mv2.

• Ex: 2000kg car moving at 10m/s has kinetic energy of 100,000J.

• Potential, U: stored energy

• Ex: One gallon of gasoline stores 138,000,000J.

Work-Energy Theorem: The net work done on an object is equal to its change in Kinetic Energy.

2212

21

ifnet mvmvW

Example: The net work done on a 20kg mass is 250J. If the mass started from rest its final speed is 5m/s: ½(20)52 – 0 = 250.

Example

• A 20kg mass is moving at 5m/s. 250J of work (net) are done on it. What is its final speed?

• A 20kg block slides across a floor. The frictional force on it is 50N. How much work is done on the block in moving 3m?

• If its initial speed was 5m/s, what is its speed after moving 3m?

• A 20kg block is pushed with 75N of force. The frictional force on it is 50N. How much work is done on the block in moving 3m?

• If its initial speed was 5m/s, what is its speed after moving 3m?

• How much work does a force perpendicular to an objects displacement do?

• Answer: Zero. The angle between F and s is 90, cos90 = 0.

The Dot Product

zzyyxx BABABABA

Example: A = (1, 1, 1), B = (5, 0, 0)

5)0)(1()0)(1()5)(1( BA

Example: Find the angle between A = (1, 1, 1) and B = (5, 0, 0)

5)0)(1()0)(1()5)(1( BA

cosABBA 3111 222222 zyx AAAA

5005 222222 zyx BBBB

cos535

7.54

3/153/5cos

FFsFW 4)0,3,4)(0,0,(

JsFW 14)0,5,2)(0,4,3(

2006 Ford MustangCurb Weight: 3450 lbs. Performance Acceleration (0-60 mph): 5.1 sec. Braking Distance (60-0 mph): 121.37 ft. Engine Type: V8 Horsepower: 300 hp

What size motor?

Cube of bricks ~ 1 ton

1 ton = 2000 lbs ~ 9000 N

Operating Speed: 10cm/s

Minimum Power:

P = Fv = (9000N)(0.1m/s)

P = 900 W = 1.2 hp

Types of Energy

• Kinetic, K energy due to motion

• Potential, U energy due to position

Some Potential Energies

• Spring: Us

• Gravitational: Ug

• Thermal: Uth

• Chemical Uch

• We use the first three of these.

Springs

• Fs = -kx, Us = ½kx2.

• k = “spring constant” in N/m and x is the change in length of the spring.

• Ex: A 100N/m spring is compressed 0.2m. It exerts (100N/m)(0.2m) = 20N of force. It stores ½(100N/m)(0.2m)2 = 2J of energy.

Gravity

• Fg = mg, Ug = mgy

• Ex: A 2kg object experiences weight (2kg)(9.8N/kg) = 19.6N. At 3m above the floor it has a stored energy of (2kg)(9.8N/kg)(3m) = 48.8Nm = 48.8J.

Conservation of Energy

• Individual energy levels change.

• Sum of all individual energies is constant.

• Change in energy is called “work”

Energy Conservation

• Total Energy E = sum of all energies

• E = K + U

• example:

• t = 0: K = 0J, U = 4000J

• later: K = 2000J, U = 2000J

Conservation of Energy

Example: Falling Ball

KE increases

U (gravitational) decreases

E = K + Ug = constant

Energy E1 E2 E3

Kinetic 0 ½mv22 0

PE-g 0 0 mgh

PE-spring

½kx2 0 0

Totals

½kx2 ½mv22 mgh

Energy E(h) E(y)

Kinetic 0 ½mv2

PE-g mgh mgy

Totals mgh ½mv2 + mgy

Energies and speeds are same at height y

Accelerations at y are not same

Energy Ei Ef

Kinetic ½mvi2 0

PE-g 0 0

Thermal 0 fks

Totals ½mvi2 fks

Example: The smaller the frictional force fk, the larger the distance, s, it will travel before stopping.

s

A 2.00kg ball is dropped from rest from a height of 1.0m above the floor. The ball rebounds to a height of 0.500m. A movie-frame type diagram of the motion is shown below.

Type E1 E2 E3 E4 E5

gravita-tional

mg(1) 0 0 0 mg(1/2)

kinetic 0 ½ m(v2)2 0 ½ m(v4)2 0

elastic 0 0 PE-elastic 0 0

thermal 0 0 PE-thermal PE-thermal PE-thermal

By energy conservation, the sum of all energies in each column is the same, = E1 = mg(1) = 19.6J

Calculate v2: (use 1st and 2nd columns)mg(1) = ½ m(v2)2.

g = ½ (v2)2.v2 = 4.43m/s

Calculate PE-thermal: (use 1st and 5th columns)mg(1) = mg(1/2) + PE-thermal

mg(1/2) = PE-thermalPE-thermal = 9.8J

Calculate PE-elastic: (use 1st and 3rd columns)PE-elastic + PE-thermal = mg(1)

PE-elastic + 9.8 = 19.6PE-elastic = 9.8J

Calculate v4: (use 1st and 4th columns)½ m(v4)2 + PE-thermal = mg(1)

½ m(v4)2 + 9.8 = 19.6½ m(v4)2 = 9.8 (v4)2 = 2(9.8)/2

v4 = 3.13m/s

Terminology

• E: total energy of a system

• E-mech = total energy minus the thermal energy

• E-mech = E – Uth.

Power: The time rate of doing work.

SI Unit: watt, W = J/s]time

workPavg

Example: How much average power is needed to accelerate a 2000kg car from rest to 20m/s in 5.0s?

work = KE 2212

21

if mvmv 2

212

21 )/0)(2000()/20)(2000( smkgsmkg

J000,400

s

J

t

workPavg 0.5

000,400 watts000,80

hpwatt

hpwatts107

746

1000,80

Horsepower: 1 hp = 746 watts

For the previous example:

avgavg vFt

sF

t

sF

t

WP )(cos)(cos

)(cos

Another equation for Power:

Ex: A car drives at 20m/s and experiences air-drag of 400N. The engine must use (400N)(20m/s) = 8,000 watts of engine power to overcome this force. 8,000 watts = 10.7 hp.

What air drag force acts at 40m/s? How much hp is needed to overcome this drag?

What size electric motor is needed to raise 2000lbs = 9000N of bricks at 10cm/s?

Minimum Power:

Pavg = Fvavg = (9000N)(0.1m/s)

P = 900 W = 1.2 hp

An object moves in a vertical circle with constant mechanical energy.

• What does this imply about its speed?

A mass on a string moves in a horizontal circle.

• Does the tension in the string vary?

• Does the tension in the string do work on the mass?

Mechanical Advantage

• F1d1 = F2d2 (E conservation)

• F2/F1 = d1/d2 = mechanical advantage

• Example: A Jack moves a car 10cm upward with fifty 20cm strokes. Mechanical advantage is 50x20/10 = 100.