Conservative Vs. Non-Conservative Forces Gravitational Potential Energy Mechanical Energy Conservation of Mechanical Energy Work

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Conservative vs. Non-conservative forces Gravitational Potential Energy

Mechanical Energy

Conservation of Mechanical energy

Work done by non-conservative forces and changes in mechanical energy

Motivation Why do we learn about potential energy and conservation of mechanical energy?

1 iClicker Quiz Work done by a gravity force

hm1  4 Mass =1 kg Mass =1 kg hm1  4 hm 3 3 vs.

hm2  2  Fmgg  hm4 1 hm4 1

(Work along h1->h2->h3->h4) vs (Work along h1->h4) Which work is greater?

A) Work along h1->h2->h3->h4 B) They are equal

C) Work along h1->h4 D) Not enough information 3

iClicker Quiz Work done by a gravity force x1=0 m x2=4 m hm1  3 Mass =1 kg

vs.  Fmgg  vs.

hm4  0

(Work along red arrows) vs (Work along green arrows) vs (work along blue arrow)

In which case, A) Work along green arrows work by gravity B) Work along blue arrow is the greatest? C) Work along red arrows D) Work along red and green are equal and greater than work along blue arrow 4 E) They are all equal

2 Mass=1 kg Can you find the total work height=10 m done by the gravity force along the path? A) Yes, of course. B) Please… No.  C) Not enough information 

height=2 m

Conservative force

If the work done by a force depends only on initial and final positions, not on the path between them, the force is called a conservative force.

Gravity force is “a” conservative force. “Spring force” is another conservative force.

3 iClicker Q: Work done by friction force

Position 1m 2m 3m 4m

- - vs. fk 1N : opposite to - displacement x1 1m x4  4m

- The works by friction in two cases are A) equal B) not equal

Is friction force a conservative force? 7

Non-Conservative force

If the work done by a force depends not only on initial and final positions, but also on the path between them, the force is called a non-conservative force.

Example: Friction force,Tension, normal force, and force applied by a person.

4 Conservative forces in Phys 102: Gravity & “Spring force (to be explained later)”

Non-Conservative forces in Phys 102: Friction, Normal force, Tension, Other applied forces

Gravitational Potential Energy

(Work done by gravity on object with mass m ) hi Wmghhg ()fi

Fmgg  mghif mgh

UUgi,, g f h f where Umghg 

U g = Gravitational “Potential Energy", which depends on position only. Energy related to position, which potentially converted to kinetic energy.

WUg gi,, U gf () U gf , U gi ,  U g Work done by gravity is negative of gravity Potential Energy change

5 Gravitational Potential Energy Mass m

Gravitational Potential Energy: Height: h

Umghg 

(Note: Height, h, should be measured along vertical direction.)

The higher and the heavier an object is, the greater the gravitational potential energy is.

initial m= 3 kg

10 m

final

30 deg iClicker Quiz The change in gravitational potential energy is ______(a)3 x 9.8 x 10 J (b) 3 x 9.8 x (-10) J (c) something else

6 initial m= 3 kg

10 m

final

30 deg Example: The change in gravitational potential energy is ______J

initial m= 3 kg

10 m

final

30 deg Example: The work done by gravity is ______

7 WUg gi,, U gf () U gf , U gi ,  U g  Work done by gravity force is negative of gravitational potential energy change

If gravitational force is the only force that does work, WWnet g

, then , from Work-Energy theorem, WK net  ,and above.

K U g  KUg 0  KU g does not change.

Conservation of Mechanical Energy with gravity force

Define Mechanical energy : 1 E KU mv2 mgh mech g 2

If WWnet g or, if gravity force is the only forces that does work,

 EEmech,, f mech i : Mechanical energy does not change.

Mechanical energy is conserved.

In general, if conservative forces are the only forces that do work, mechanical energy is conserved.: “Conservation of mechanical energy”

8 Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy

can change, but their sum, the mechanical energy Emec of the system, cannot change.

Example: Falling ball Less kinetic energy, but h More potential energy 1 v1 Height

More kinetic energy, but Less potential energy h2 v2

Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy

can change, but their sum, the mechanical energy Emec of the system, cannot change.

Example: Falling ball Mechanical energy h 1 1 v1 2 Emvmghmech  Height 2

Conservation of mechanical energy 11 mv22 mgh mv  mgh h2 v2 2211 22

9 19

iClicker Quiz A person throws a ball 30 degree from horizontal from the top of a 20 m high building at the speed of 5 m/s. Neglect the air resistance. True or false? If he throws the ball at a different angle but with the same speed, the ball would hit the ground at a different speed. (a) True (b) False

Example: Find the speed when the ball hits the ground.

10 Example: Pendulum

2 m rope

30 degree V_i=0

Air resistance is negligible. What is the speed at the bottom?

Spring force : Conservative force Force from spring on object attached to the spring Direction: toward equilibrium point Magnitude: proportional to the distance from equilibrium point Hooke’s law: x: displacement from Fxkxspring () relaxed position

k: spring constant (N/m)

Slope= -k

0 x

For x negative, F positive 22

11 iClicker Quiz

You need 4 N force to stretch a spring by 0.1 m. How large force would you need to stretch the same spring 0.2 m? a) 2 N b) 4 N c) 8 N d) 16 N

Example For the same spring, what is spring constant?

When the spring is compressed by 0.2m, what is the force from the spring?

Spring Potential Energy

Spring (elastic) potential energy : 1 UxUkx() 2 spring s 2

x: displacement from relaxed (equilibrium) position k: spring constant (N/m)

Either compressed or stretched spring has a positive potential energy.

(For derivation, see textbook.)

12 iClicker Quiz A spring stretched by 0.1 m has 1 J spring potential energy. What would be the spring potential energy if the same spring is compressed by 0.2 m?

a) 2 J b) 4 J c) -2 J d) -4 J Example: What is the spring constant?

Work done by spring force & Spring Potential Energy change Just like work done by gravity force & gravitational potential energy change Work done by spring force is negative of spring P.E. change

WUUUs ssfsi (),, 

, where spring (elastic) potential energy : 1 UxUkx() 2 spring s 2

13 Conservation of Mechanical Energy with gravity and spring Work-Energy Theorem: K KKWf inet 

Define Mechanical energy : 11 E KU  Umv22 mgh kx mech g s 22

If WWWnet g s

or, if gravity and spring are the only forces that do work,

 KWg  Wsgs  U U

KUgs U0 Emech 0

 EEmech,, f mech i : Mechanical energy does not change. If conservative forces are the only forces that do work, mechanical energy is conserved.: “Conservation of mechanical energy” 27

Conservation of Mechanical Energy for Spring 1 Elastic (or, Spring) Potential Energy: Uxkx() 2 elastic 2 11 Mechanical energy: Emvkx22 mech 22 Conservation of mechanical energy 1111 mv22 kx mv 22  kx 222211 22

1 K = 0

U  K

U = 0 2

14 Example: Spring potential A block of mass m = 0.40 kg slides across a horizontal frictionless counter with a speed of v = 0.50 m/s. It runs into and compresses a spring of spring constant k = 750 N/m. When the block is momentarily stopped by the spring, by what distance d is the spring compressed?

Work done on a system by non-conservative force

If non-conservative forces do work on an object, in addition to conservative forces,

 Mechanical energy changes by the amount of work done by the non-conservative force.

EEmech mech,, f  E mech i  W non conservative

Emech  K U

15 Example: A skier stars from rest at the top of a frictionless incline of height 20 m . At the bottom of the incline, the skier encounters a horizontal surface where the  k =0.21 . How far does the skier travel on the horizontal surface before coming to rest?

iClicker 1: In this problem, work done by normal force is _____ .

iClicker 3: In this problem, work done by friction force is _____ . (a) positive, (b)zero, (c)negative

A 12-kg projectile is launched with an initial vertical speed of 20 m/s. It rises to a maximum height of 18 m above the launch point. How much work is done by the dissipative (air) resistive force, which is a non- conservative force, on the projectile during this ascent?

16 WFx PFv (in 1D) tt

In 2D, Instantaneous Power:    FvcosF ,v

Work done by a constant force     Force W  F d cosF ,d  F d 

Displacement

Instantaneous Power by a force     Force P  F v cosF ,v  F v 

Velocity

17 A weightlifter, is able to lift 250 kg 2.00 m in 2.00 s. What is his power output?

A jet engine develops 1.0 x 10^5 N of thrust in moving an airplane forward at a speed of 250 m/s. What is the power developed by the engine?