Chapter 4Work and Energy

Additional Concepts For Describing Motion

PSC 150 Exercise

"Conservation of Energy"

Results and Conclusions

mass = 10kgh, m v, m/s KE, J GPE, J E, J

Free-fall

20.86 34.5164.63 18.2381.48 1.3673.53 -12.5639.16 -28.84

1.5 -39.63

5955 20441662 6334

9 7985789 7206

4159 38387853 147

800080007990800080008000

1.) Based on your table, as the object moves UPWARD its kinetic energy:

Questions:

Decreases

2.) Based on your table, as the object moves UPWARD its gravitational potential energy:

3.) Based on your table, as the object moves DOWNWARD its kinetic energy:

4.) Based on your table, as the object moves DOWNWARD its gravitational potential energy:

5.) Based on your table, as the object moves UPWARD its total mechanical energy:

6.) Based on your table, as the object moves DOWNWARD its total mechanical energy:

7.) As the object moves the only force acting on it is:

Decreases

Increases

Increases

Remains Constant

Remains Constant

Gravitational Force

If the only force acting on an object is the gravitational force the kinetic and gravitational potential energies may change but the total mechanical

energy remains constant.

Conclusion

mass = 5kgh, m v, m/s KE, J GPE, J E, J

Pendulum

4.64 5.711.73 9.160.15 10.711.52 9.383.02 7.655.91 1.36

82 227210 85287 7220 74146 148

5 290

309295294294294295

8.) Based on your table, as the pendulum swings DOWNWARD its kinetic energy:

Questions

9.) Based on your table, as the pendulum swings DOWNWARD its gravitational potential energy:

10.) Based on your table, as the pendulum swings UPWARD its kinetic energy:

11.) Based on your table, as the pendulum swings UPWARD its gravitational potential energy:

12.) Based on your table, as the pendulum swings UPWARD or DOWNWARD its total mechanical energy:

13.) As the pendulum swings the two forces acting on it are:

Increases

Increases

Decreases

Decreases

Remains Constant

Gravitational Force and Tension

Tension is always perpendicular to the direction of motion.

Conclusion

If the only force acting on an object is the gravitational force or if there are other forces which are always perpendicular to the direction of

motion the kinetic and gravitational potential energies may change but

the total mechanical energy remains constant.

A

B

C

D

Equilibrium Level

Highest Point

Highest Point

Pendulum

14.) At what point(s) (A,B,C,D) does the kinetic energy have its maximum value?

15.) At what point(s) (A,B,C,D) does the gravitational potential energy have its maximum value?

16.At what point(s) (A,B,C,D) does the pendulum have both kinetic and potential energy?

C

A & D

B

mass = 5 F = 20h. m v, m/s KE, J GPE, J E, J Δ , E J , d m F x d, J %Δ

Horizontal Force

0 50 10.10 16.20 21.30 23.60 25.8

63 0255 0656 0

1134 01392 01664 0

63255656

113413921664

***192593

107113291601

09.729.753.766.380.1

***194594

107413261602

***1.0%0.2%0.3%-0.2%0.1%

kg N

Questions:

17.) Based on your table, as the object moves its kinetic energy:

18.) Based on your table, as the object moves its gravitational potential energy:

19.) Based on your table, as the object moves its total mechanical energy:

20.) As the object moves the applied force was in the SAME OPPOSITE direction as the motion.

21.) The units of Force X Displacement are:

22.) The units of Force X Displacement are the same as the units of mechanical energy:

23. As the object moves the change in its total mechanical energy, E, approximately equals the product of the applied force and the displacement.

Increases

Remains Constant

Increases

Same

kg⋅m2

s2

True

True

ConclusionWhen a net external force acts on an object in the same direction as

its motion the total mechanical energy increases.

The change in the object’s total mechanical energy equals the

product of the net force and the object’s displacement.

mass = 3kg f = -20Nh, m v, m/s KE, J GPE, J E, J ΔΕ, J , d m F x d, J %Δ

Frictional Force

0 330 28.70 23.60 13.60 9.30 1.3

019.839.967.875.281.6

1634 01236 0835 0277 0130 03 0

163412368352771303

***-398-799

-1357-1504-1631

***-396-798

-1356-1504-1632

***-0.8%-0.2%-0.1%0.0%0.1%

Questions:

17.) Based on your table, as the object moves its kinetic energy:

18.) Based on your table, as the object moves its gravitational potential energy:

19.) Based on your table, as the object moves its total mechanical energy:

20.) As the object moves the frictional force was in the SAME OPPOSITE direction as the motion.

23. As the object moves the change in its total mechanical energy, E, approximately equals the product of the applied force and the displacement.

Decreases

Remains Constant

Decreases

Opposite

True

ConclusionWhen a frictional force acts on an object in the opposite direction as

its motion the total mechanical energy decreases.

The change in the object’s total mechanical energy equals the product of the frictional force and the object’s displacement.

Define :r F ⋅

r d≡work

Units :

kg⋅m2

s2= ,Joule J

If r F &

r d are in opposite directions

the work is negative and

the work is done BY the object.

If r F &

r d are in the same direction

the work is positive and the work

is done ON the object.

Work is done when a force acts

on an object AND the object moves

parallel to that force. Work is a scalar!

Define :

Kinetic energy is something an

object has because it is moving.

kinetic energy, KE ≡12 mv2

Units :

kg⋅m2

s2= ,Joule J

KE is a scalar!Change in kinetic energy,

ΔKE =KE f -KE i =12 mvf

2 −12 mvi

2

If Δ ,KE is positive the kinetic energy hasincreased.

If Δ ,KE is negative the kinetic energy hasdecreased.

Work-Energy Theorem“When a net force causes an object to accelerate, the work done on or by the object equals the change in the object’s kinetic energy.”

The Work-Energy Theorem can be derived from Newton’s Second Law.

Wnet force =ΔKE

r F =m

r a

multiply both sides by "r d "

r F ⋅

r d=m

r a⋅ r d

Using : vf2 = vi

2 + 2ad

r a ⋅

r d=

vf2 −vi

2

2Substituting gives :

r F ⋅

r d=m

vf2 −vi

2

2 ⎛

⎝ ⎜

⎞

⎠ ⎟

Simplyfing gives :

r F ⋅

r d=1

2 mvf2 −1

2 mvi2 ⇒ W = ΔKE

Another Type of EnergyDefine:Gravitational Potential Energy is something an object has because of its position.

Gravitational Potential Energy, GPE≡mgh

The operational definition of GPE

requires the choice of a "reference level"

from which "h" can be measured.

The choice is arbitrary but is usually

chosen at the object's initial position.

GPE is a Scalar!

Units :

kg⋅m2

s2= ,Joule J

GPE depends on where the object is.

Let the floor be the reference level

h=0

h=.6mh=1.1mh=2.2m

change in gravitational potential energy,

ΔGPE =mghf −m ghi

“When an external force (equal to the object’s weight) lifts an object at a constant velocity, the work done by that force equals the change in the object’s Gravitational Potential Energy.”

Wconstantvelocity =ΔGPE

If ΔGPE is positive, the gravitational Potential Energy has increased.

If ΔGPE is negative, the gravitational Potential Energy has decreased.

Total Mechanical Energy, EThe Total Mechanical Energy of an object is defined as the sum of its Kinetic Energy and Gravitational Potential Energy.

E = KE + GPE Extended Work-Energy Theorem

“The work done by any force other than the gravitational force equals the change in Total Mechanical Energy.” WF≠FG

=ΔE

What if the only force acting on an object is the gravitational force?

Work done by the gravitational force does NOT change the total mechanical energy it does cause a conversion between kinetic energy and gravitational potential energy.

If the work done by the gravitational force is positive…the gravitational force is in the same direction as the displacement, gravitational potential energy is converted into kinetic energy.

The gravitational force is called a Conservative Force.

WFG

(+)L GPE → KE

If the work done by the gravitational force is negative…the gravitational force is in the opposite direction as the displacement, kinetic energy is converted into gravitational potential energy.

WFG

(−)L KE → GPE

The Law of Conservation of Energy“If the only force acting on an object is the gravitational force, or if there are other forces acting on the object but they do no work, the kinetic and gravitational energies may change but the total mechanical energy remains constant.”

Wnonconservative =0⇒ ΔE =0

General Work-Energy Theorem

Kinetic Energy

GravitationalPotential Energy

Net Force

Lifting Forcegreater than

weight

Work Done On Object BY

Non-Conservative Forces

Work Done BY Object Against

Non-Conservative Forces

e.g.,Friction Lifting ForceLess than

weight

Work Done by

Conservative Forcese.g., gravity

Total Energy Increases

Total Energy Decreases

Total Energy Constant