PHY 2311

PHYSICS 231Lecture 11+12: How much energy

goes into problems?

Remco Zegers

PHY 2312

Previously Work: W=Fcos()x Energy transfer Power: P=W/t Rate of energy transfer Potential energy (PE) Energy associated with

position. Gravitational PE: mgh Energy associated with

position in grav. field. Kinetic energy KE: ½mv2 Energy associated with

motion

NEXT: Conservative force: Work done does not depend on path Non-conservative force: Work done does depend on path Mechanical energy ME: ME=KE+PE

Conserved if only conservative forces are present KEi+PEi=KEf+PEf

Not conserved in the presence of non-conservative forces (KEi+PEi)-(KEf+PEf)=Wnc

PHY 2313

quiz (for credit)

A person is dragging a block overa floor, with a force parallel tothe floor.After 4 meter, the floor turnsrough and instead of a force of 2N and force of 4N must be applied.The force-distance diagram shows to situation.How much work did the person do over 8 meter?a) 0 J b) 16 J c) 20 J d) 24 J e) 32 J

4 8 m

2N

4N

0

force

distance

Work: area under F-x diagram: 4x2+4x4=24 J

PHY 2314

Work and energy

WORK

POTENTIAL ENERGY

KINETIC ENERGY

PHY 2315

Mechanical Energy

Mechanical energy

GravitationalPotential

Energy (mgh)

Kinetic Energy½mv2

PHY 2316

Conservative forces

A force is conservative if the work done by the force whenMoving an object from A to B does not depend on the pathtaken from A to B.

Example: gravitational force

h=

10

m

Using the stairs:Wg=mghf-mghi=mg(hf-hi)

Using the elevator:Wg=mghf-mghi=mg(hf-hi)

The path taken (longer or shorter)does not matter: only the displacement does!

PHY 2317

Non conservative forces

A force is non-conservative if the work done by the force when moving an object from A to B depends on the path taken from A to B.

Example: FrictionYou have to perform more work Against friction if you take the long path, compared to the shortpath. The friction force changeskinetic energy into heat.

Heat, chemical energy (e.g battery or fuel in an engine)Are sources or sinks of internal energy.

PHY 2318

Conservation of mechanical energy only holds if the system is closed AND all forces are conservative

MEi-MEf=(PE+KE)i-(PE+KE)f=0 if all forces are conservativeExample: throwing a snowball from a building neglecting air resistance

MEi-MEf=(PE+KE)i-(PE+KE)f=Wnc if some forces are nonconservative.

Wnc=work done by non-conservative forces.

Example: throwing a snowball from a building taking into account air resistance

PHY 2319

Overview

Newton’s second LawF=ma

WorkW=(Fcos)x

Equations of kinematicsX(t)=X(0)+V(0)t+½at2

V(t)=V(0)+at

Work-energy TheoremWnc=Ef-Ei

Conservation of mechanicalenergy

Wnc=0 Closed system

PHY 23110

Conservation of mechanical energyA) what is the speed of m1 and m2 when they pass each other?

(PE1+PE2+KE1+KE2)=constantAt time of release:PE1=m1gh1=5.00*9.81*4.00 =196. JPE2=m2gh2=3.00*9.81*0.00 =0.00 JKE1=0.5*m1*v2=0.5*5.00*(0.)2 =0.00 JKE2=0.5*m1*v2=0.5*3.00*(0.)2 =0.00 JTotal =196. JAt time of passing:PE1=m1gh1=5.00*9.81*2.00 =98.0 JPE2=m2gh2=3.00*9.81*0.00 =58.8 JKE1=0.5*m1*v2=0.5*5.00*(v)2 =2.5v2 JKE2=0.5*m1*v2=0.5*3.00*(v)2 =1.5v2 JTotal =156.8+4.0v2196=156.8+4.0v2 so v=3.13 m/s

PHY 23111

work

How much work is done by thegravitational force when the massespass each other?

W=Fx=m1g2.00+m2g(-2.00)=39.2 J

Pestart- Pepassing=(196.-98.-58.8)=39.2 J

The work done by Fg is the same asthe change in potential energy

PHY 23112

Friction (non-conservative)The pulley is not completely frictionless.The friction force equals 5 N. What is thespeed of the objects when they pass?

(PE+ KE)start-(PE+KE)passing=Wnc

Wnc=Ffrictionx=5.00*2.00=10.0 J

(196.)-(156.8+KE)=10 JKE=29.2 J=0.5*(m1+m2)v2=4v2

v=2.7 m/s

PHY 23113

A spring

Fs=-kx

+xFs(x=0)=0 NFs(x=-a)=kaFs=(0+ka)/2=ka/2

Ws=Fsx=(ka/2)*(a)=ka2/2The energy stored in a spring depends on the location of the endpoint: elastic potential energy.

k: spring constant (N/m)

PHY 23114

PINBALLThe ball-launcher spring has aconstant k=120 N/m. A playerpulls the handle 0.05 m. The mass of the ball is 0.1 kg. Whatis the launching speed?

(PEgravity+PEspring+KEball)pull=(PEgravity+PEspring+KEball)launch

mghpull+½kxpull2+½mvpull

2 = mghlaunch+½kxlaunch2+½mvlaunch

2

0.1*9.81*0+½120(0.05)2+½0.1(0)2=0.1*9.81*(0.05*sin(10o))+½120*(0)2+½0.1vpull

2

0.15=8.5E-03+0.05v2

v=1.7 m/s

PHY 23115

Ball on a trackA

B

In which case has the ball the highest velocity at the end?A) Case A B) Case B C) Same speed

In which case does it take the longest time to get to the end?A) Case A B) Case B C) Same time

h

h

end

end

PHY 23116

Race track

KE PE TME NC KE PE TME NC KE PE TME NC KE PE TME NC KE PE TME NCWith friction

PHY 23117

PHY 23118

A swing

h

30o

L=5m

If relieved from rest, what isthe velocity of the ball at the lowest point?

(PE+KE)=constantPErelease=mgh (h=5-5cos(30o)) =6.57m JKErelease=0

PEbottom=0KEbottom=½mv2

½mv2=6.57m so v=3.6 m/s

PHY 23119

PHY 23120

A running personWhile running, a person dissipates about0.60 J of mechanical energy per step perkg of body mass. If a 60 kg person developsa power of 70 Watt during a race, howfast is she running (1 step=1.5 m long)What is the force the person exerts on theroad?

W=Fx P=W/t=FvWork per step: 0.60 J/kg * 60 kg=36 JWork during race: 36*(racelength(L)/steplength)=24LPower= W/t=24L/t=24vaverage=70 so vaverage=2.9 m/sF=P/v so F=24 N

PHY 23121

Parabolic motion

t=0 t=3t=2t=1

t=5

Where is the kinetic energy…1) highest?2) lowest ?Assume height of catapult isnegligible to the maximum heightof the stone.

AE

DCB

And what about potential energy?

PHY 23122

PHY 23123

question

An object is lowered into a deep hole in the surface ofthe earth. What happens to its potential energy?

a) increaseb) decreasec) remain constantd) cannot tell from information givene) don’t know

PHY 23124

question

An outfielder throws a baseball of 0.15 kg at a speed of40 m/s and angle of 30 degrees with the field. What is thekinetic energy of the baseball at the highest point,ignoring friction?

a) 0 Jb) 30 Jc) 90 Jd) 120 Je) don’t know

Horizontal component of velocity at start:vx=vocos30=34.65 m/s

At highest point: only horizontal velocityvx,highest=34.65 m/skinetic energy: 0.5mv2=90 J

PHY 23125

question

A worker pushes a sled with a force of 50 N overa distance of 10 m. A frictional force acts on thewheelbarrow in the opposite direction, with a magnitude of 30 N. What net work is done on thewheelbarrow?a) don’t knowb) 100 Jc) 200 Jd) 300 Je) 500 J

Wfriction=Fx=(50-30)10=200 J

PHY 23126

question

Old faithful geyser in Yellowstone park shoots water hourlyto a height of 40 m. With what velocity does the water leave the ground?a) 7.0 m/sb) 14 m/sc) 20 m/sd) 28 m/se) don’t know

At ground level:E=0.5mv2+mgh= 0.5mv2+0=0.5mv2

At highest point:E=0.5mv2+mgh= 0+m*9.8*40=392mConservation of energy:0.5mv2=392m so v=28 m/s

PHY 23127

quiz (for credit)

A ball of 1 kg rolls up a ramp, with initial velocity of 6 m/s.It reaches a maximum height of 1 m (I.e. velocity 0 atat that point). How much work is done by friction?

a) 0.b) 8.2 Jc) 9.8 Jd) 18 Je) 27.8 J

initial: E=0.5mv2 (kinetic only)=18Jfinal: E=mgh (potential only)=9.8 JWnc=18-9.8=8.2 J

kinetic energy: 0.5mv2

potential energy: mgh g=9.8 m/s2