Work and Energy

Work

The work done by a constant force is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement.

dFW cos

dFW

W = Work units of J or Nm or The Joule is named after James Prescott Joule

is the angle between the force and the displacement

2

2

s

mkg

Note that Fcos is the component of the force in the direction of the displacement. If the angle is greater than ninety degrees then the work will be negative (cos.

F

d

F

cosF

Work is a scalar quantity. Energy is defined as the ability to do work and therefore is a scalar quantity as well. Work can be positive or negative but these signs are not direction. We will see that they indicate a gain of kinetic energy or a loss of kinetic energy respectively.

Negative work is done on an object when it is slowed by a force. Positive work is done when an object is sped up by a force.

The area under a F-d graph is equal to the work done by an applied force. Assume the force and displacement are co-linear.

The total work done on an object is the sum of all the work done by individual forces.

dFW

dFW R

a) Rousseau pushes with a force of 500 N on an immovable wall. How much work is done on the wall?

b) David swings a rock around his head with a centripetal force of 250 N. The rock goes around his head 3 times in 0.56 s (the radius of the circle is 0.8 m). What is the work done on the rock?

c) A 4 kg block is raised 5 m. How much work is done on the block if one assumes it was lifted with a constant velocity?

aF

gF

workF a

JW

ms

mkgW

dmgW

dFW g

2.196

)1)(5)(81.9)(4(

0cos

cos

2

JW

mkg

NkgW

dmgW

dFW g

2.196

)1)(5)(81.9)(4(

180cos

cos

workF g

This means the work done on the ball is 0 J.

N.B.

JW

W

dFW net

0

)5)(0(

d) A sled (15 kg) is pulled with a 50 N [20o ath] force for 7.5 m. The coefficient of friction is 0.21. Calculate the work done by each force and total work done on the sled.

NF

aF

gF

fF

20o

Work done by FN and Fg are zero since they are perpendicular to the displacement.

Work done by Fa

JW

mNW

dFW a

4.352

)20)(cos5.7)(50(

cos

][0.130

)81.9)(15(20sin50

0

upNF

kg

NkgF

FFF

FFF

N

N

gayN

gayN

To calculate work done by friction we must calculate FN.

Work done by Ff

JW

W

dFW

dFW

N

f

8.204

)1)(5.7)(0.130)(21.0(

180cos

cos

Therefore the total work done on the sled is 147.6 J

WORK ENERGY THEOREM

For an object that is accelerated by a constant net force and

moves in the same direction . . .

cos)(

cos

cos

12 dt

vvmW

dmaW

dFW net

22

)(2

)2

)((

)1()(

21

22

21

22

1212

12

mvmvW

vvm

W

vvvvmW

t

dvvmW

define kinetic energy as (Ek)

2

2mv

k

kk

EW

EEW

12

The work done by the net force acting on a body is equal to the change in the kinetic energy of the body.

22cos

21

22 mvmv

dF

A ball is dropped from rest at a height of 8.25 m. What will be its speed when it hits the ground? (could solve this kinematically but let’s do it using the work-energy theorem)

22cos

22cos

22

22

12

12

mvmvdmg

mvmvdF

EW

g

k

gF

s

mv

vm

s

m

72.12

02

)1)(25.8)(81.9(

2

2

2

2

Therefore the speed of the ball is 12.72 m/s when it hits

the ground.

GRAVITATIONAL POTENTIAL ENERGY

Conservative and Non-Conservative Forces

A ball is thrown upwards and returns to the thrower with the same speed it departed with.

A block slides into a spring, compresses it and leaves the spring with the same speed it first contacted it with.

A force is conservative if the kinetic energy of a particle returns to its initial value after a round trip (during the trip the Ek may vary). A force is non-conservative if the kinetic energy of the particle changes after the round trip (Assume only one force does work on the object). Gravitational, electrostatic and spring forces are conservative forces. Friction is an example of a non-conservative force. For a round trip the frictional force generally opposes motion and only leads to a decrease in kinetic energy.

We must introduce the concept of potential energy. This is energy of configuration or position. As kinetic energy decreases the energy of configuration increases and vice versa.

Ep Change in potential energy

Eg Change in gravitational potential energy

Ee Change in elastic potential energy

0 pk EE

gk EE

)1(

)1(

cos

dmgE

or

dmgE

dFE

WE

g

g

g

g

hmgEg

define Eg as mgh

h is height relative to a reference point

Gravity does work on an object as its height changes. As an object increases its height gravity does negative work on the object and the object’s kinetic energy decreases. This loss of kinetic energy is a gain of potential energy.

2

22

1

21

12

21

22

22

0)()22

(

mghmv

mghmv

mghmghmvmv

21 mm EE

define Em as the mechanical energy

Mechanical Energy is conserved when an object is acted upon by conservative forces.

LAW OF CONSERVATION OF ENERGY

Energy may be transformed from one kind to another, but it cannot be created or destroyed: the total energy is constant. There are many forms of energy

such as electromagnetic, electrical, chemical, nuclear, and thermal.

0.... dcba EEEE

a) A ball is launched from a height of 2 m with an initial velocity of 25 m/s [35o ath]. What is the speed of the ball when its height is 7.5 m?

2

22

1

21

2

22

1

21

22

22

ghv

ghv

mghmv

mghmv

s

mv

ms

mm

s

m

s

mv

ghghvv

73.22

)5.7)(81.9(2)2)(81.9(2)25(

22

2

2222

2

2121

22

The speed of the ball is 22.73 m/s

The energy approach doesn’t calculate the velocity but it is quicker. The kinematics approach is longer but more precise.

ELASTIC POTENTIAL ENERGY

Hooke’s Law (Robert Hooke 1678)

The magnitude of the force exerted by a spring is directly proportional to the distance the spring has moved from its equilibrium position. An ideal spring obeys Hooke’s Law because it experiences no internal or external friction.

x (m)

F (N)slope = k

elastic limit

non-elastic region

breaking point

The linear region is sometimes called Hooke’s Law region. It applies to many elastic devices.

Hooke’s Law

F = force exerted on the spring (N)

k = force constant of spring (N/m)

x = position of spring relative to the equilibrium (deformation) (m)

The direction of compression on the spring is negative while the direction of elongation is positive (for F and x). The spring exerts an equal and opposite force on the object.

Derivation of Elastic Potential Energy

A spring exerts a conservative force on a object. An object will have the same kinetic energy after a round trip with a spring. The spring will begin at its equilibrium position with zero potential energy.

kxF

WE

EE

EE

e

ke

pk

0

2

)1(2

cos)(

2

2

2

12

kxE

xkx

E

dFEE

e

e

ee

zero since at equilibrium

the force on the object and its d have opposite directions

For an object interacting with an ideal spring.

2

2

1kxEkxF e

The potential energy of this object must be considered in the mechanical energy.

2222

22

2

221

1

221 kxmgh

mvkxmgh

mv

Remember a reference height is needed for height. The direction of x is not important unless solving for x. If it is known that the answer is compression then –x is correct. If the answer is elongation then +x is correct.

If one form of energy is not present then it need not be included in the equation.

a) A 2 kg ball is dropped from a height of 10 m onto a spring that is 0.75 m in length and has a spring constant of 1000 N/m. How far will the ball compress the spring? What force is exerted on the ball at its lowest point?

10 m

0.75 m x2 0.75+x2

initialv1=0h1=10 mx1=0

finalv2=0h2=0.75+x2

2)75.0(

2

2222

22

21

22

21

22

2

221

1

221

kxxmgmgh

kxmghmgh

kxmgh

mvkxmgh

mv

Use the quadratic formula to solve.

mcompressesspringthe

mxormx

6224.0

6224.05832.0 22

][4.622

)6224.0)(1000(

...

upNF

mm

NF

kxF

bycalculatedisthisatballtheonspringthebyexertedforceThe