Work & Energy

Energy is Conserved

• Energy is “Conserved” meaning it can not be created nor destroyed – Can change form– Can be transferred

• Total Energy does not change with time.

•This is a BIG deal!

Energy

• Forms– Kinetic Energy Motion (Today)– Potential Energy Stored

(Wednesday)– Heat later– Mass (E=mc2) 1161

• Units Joules = kg m2 / s2

The definition of work, when the force is parallel to the displacement:

(7-1)

SI work unit: newton-meter (N·m) = joule, J

Work

Typical Work

Work done may be positive, zero, or negative, depending on the angle between the force and the displacement:

Positive & Negative Work

If the force is at an angle to the displacement: (7-3)

Only the horizontal component of the force does any work (horizontal displacement).

Force at an Angle Work

Work by Constant Force at Angle

• You pull a 30 N chest 5 meters across the floor at a constant speed by applying a force of 50 N at an angle of 30 degrees. How much work is done by the 50 N force?

30

50 N

W = T cos xcos (30) m) = 217 Joules

T

mg

N

f

Tx

Ty

Where did the energy go?• You pull a 30 N chest 5 meters across the floor at

a constant speed, by applying a force of 50 N at an angle of 30o.

• How much work did gravity do?

• How much work did friction do?mg

90

xW = Fcos x30 cos(90) x 5 T

mg

N

fX-Direction: F = ma T cos(30) – f = 0 f = T cos(30) = 43.3 N

f r

180

W = f cos(x43.3 N) cos(180) x 5Joules

Perpendicular Force and Work

A car is traveling on a curved highway. The force due to friction fs points toward the center of the circular path.

How much work does the frictional force do on the car?

Zero!

General Result: A force that is everywhere perpendicular to the motion does no work.

Work Done by GravityIn lifting an object of weight mg by a height h, the person doing the lifting does an amount of work W = mgh.

If the object is subsequently allowed to fall a distance h, gravity does work W = mgh on the object.

W mgh

When positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

Work By Gravity

Kinetic Energy: Motion• Apply constant net force along x-

direction to an object m.W = Fx x = m a x = ½ m (v2 – v0

2)

• Work changes ½ m v2 • Voila – new concept:• Define Kinetic Energy K = ½ m v2

W = K

2 20 0

2 20

2 ( )

1

2

v v a x x

a x v v

Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy.

(7-7)

Work-Energy Theorem

Falling Ball Example A ball falls a distance 5 meters, what is final

speed?

Only force/work done by gravityW = KEWg = ½ m(vf

2 – vi2)

Fg h = ½m vf2

mgh = ½m vf2

Vf = sqrt( 2 g h ) = 10 m/s

mg

Work: Energy Transfer due to Force

• Force to lift trunk at constant speed– Case a Ta – mg = 0 T = mg– Case b 2Tb - mg =0 or T = ½ mg

• But in case b, trunk only moves ½ distance you pull rope.

• F * distance is same in both!Ta

mg

Tb

mg

Tb

Loading with a CraneA 3,000 kg truck is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force, which is large enough to overcome the gravitational force and keep the truck moving upward, is applied over a distance of 2.0 m.

(a) Find the work done on the truck by the crane.

(b) Find the work done on the truck by gravity.

(c) Find the net work done on the truck.app app (31 kN)(2.0 m) 62 kJyW F y

2g (3000 kg)( 9.81 m/s )(2.0 m) 58.9 kJyW mg y

net app g (62.0 kJ) ( 58.9 kJ) 3.1 kJW W W

Preflight 1 You are towing a car up a hill with constant velocity. The work done on the car by the normal force is:

1. positive 2. negative 3. zero

T

W

FN V

“work done by normal force is 0 because q is 90 and cos 90 = 0”

correct

28

38%13%50%

Preflight 2 You are towing a car up a hill with constant velocity. The work done on the car by the gravitational force is:

1. positive 2. negative 3. zero

The work done on the car by the gravitational force is negative because it hinders motion up the hill

W

T

FN V

correct

0%63%37%

Preflight 3 You are towing a car up a hill with constant velocity. The work done on the car by the tension force is:

1. positive 2. negative 3. zero

“work done by tension is positive because the angle of the force is between 0 and 90”

W

T

FN V

correct63%25%12%

KE ACTS

b) 0.707 va) 0.50 v e) 2.00 vd) 1.414 vc) v

Car 1 has twice the mass of Car 2, but they both have the same kinetic energy. If the speed of Car 1 is v, approximately what is the speed of Car 2?

2 21 1 2 2

1 2

2 22 1 2 2

2 21 2

2 1

1 1

2 22

2

2

2

m v m v

m m

m v m v

v v

v v

Preflight 4 You are towing a car up a hill with constant velocity. The total work done on the car by all forces is:

1. positive 2. negative 3. zero

The total work done is positive because the car is moving up the hill. (Not quite!)

Total work done on the car is zero because the forces cancel each other out.

W

T

FN V

correct

63%12%25%

Block w/ friction A block is sliding on a surface with an initial speed of

5 m/s. If the coefficent of kinetic friction between the block and table is 0.4, how far does the block travel before stopping?

5 m/s

mg

N

f x

y

Y direction: F=ma N-mg = 0 N = mg

Work WN = 0 Wmg = 0 Wf = f x cos(180) = -mg x

W = K -mg x = ½ m (vf

2 – v02)

-g x = ½ (0 – v02)

g x = ½ v02

x = ½ v02 / g

= 3.1 meters

Summary• Energy is Conserved• Work = transfer of energy using

force – Can be positive, negative or zero– W = F d cos()

• Kinetic Energy (Motion)– K = ½ m v2

• Work = Change in Kinetic Energy W = K