Energy Conservation

And Non-Conservative Forces

Conservation of Mechanical EnergyConservation of Mechanical EnergyDefinition of mechanical energy:

(8-6)

Using this definition and considering only conservative forces, we find

Or equivalently

Conservation of Mechanical EnergyConservation of Mechanical EnergyEnergy conservation can make kinematics problems much easier to solve

Catching a Home RunAt the bottom of the 9th inning,a player hits a 0.15 kg baseball overthe outfield fence. The ball leaves the bat with a speed of 36.0 m/sand a fan in the bleachers catchesit 7.2 m above the point where itwas hit. Neglect air resistance.

(a) What is the kinetic energy Kf of the ball when caught?

(b) What is the speed vf of the ball when caught.1 12 22 2

0 (0.15 kg)(36 m/s) 97 Ji i i i iE U K mgy mv

2(0.15 kg)(9.81 m/s )(7.2 m) 11 Jf fU mgy

97 11 J

86

i f

f

f

E E

J K

K J

2186

2

17233.9 /

0.15

f

f

mv J

Jv m s

Kg

Speed and PathEnergy is a scalar. The speed of the cap is vi at height yi and its speed is vf at height yf, independent of the path between the two heights.Thus the angle at which the cap is launched does not change this result, as long as vi is large enough to carry the cap to height yf.

Energy of a Ball ACT

2m

m

h

When a ball of mass m is dropped from height h, its kinetic energy just before landing is K.If a 2nd ball of mass 2m is dropped from height h/2, what is its kinetic energy just before landing?

(a) K/4 (b) K/2 (c) K (d) 2K (e) 4K

Non-conservative ForcesNon-conservative ForcesIn the presence of non-conservative forces, the work done by these forces must be included in the energy calculations.

_ _initial finalTotal Energy Total Energy

1 1 2 2 ncU K U K E

This term is equal to the

energy expended in

overcoming the non-

conservative force (usually

friction) or to the energy

gained from work done by

the non-conservative force.

Enc = -Wnc

Galileo’s Pendulum ACTHow high will the pendulum swing on the other

side now?

A) h1 > h2 B) h1 = h2 C) h1 < h2

h1 h2

m

Conservation of Energy (Wnc=0)

Kinitial + Uinitial = Kfinal+Ufinal

0 + mgh1 = 0 + mgh2

h1 = h2

44

Find the Diver’s DepthFind the Diver’s Depth A 95.0 kg diver steps off a diving board and drops into the water, 3.00 m below. At some depth d below the water’s surface, the diver comes to rest.

If the non-conservative work done on the diver is Wnc = -5,120 J, what is the depth d?

i i iE U K

f f f ncE U K E

f i

f nc i

E E

U E U

0 of U

0

0

( 5120 ) 5120nc ncE W J J

2 2(95.0 ) 9.8 ( ) 5120 (95.0 ) 9.8 (3.00 )m mkg d J kg ms s

2793 51202.5

931

J Jd m

N

Judging a PuttJudging a PuttA golfer badly misjudges a putt, giving the ball an initial speed v1, which sends the ball a distance d that is only one quarter of the distance to the hole.

If the non-conservative force f due to the resistance of the grass is constant, what initial speed v2 would have been needed to putt the ball from its initial position to the hole?

1 2

21

1

2

E E

mv fd

22

1(4 )

2mv f d

2 12v v

1

22

14

2mv fd

2 22 1

2 22 1

1 14

2 2

4

mv mv

v v

Landing with a ThudLanding with a Thud A block of mass m1 = 2.40 kg is on a horizontal table with a coefficient of friction k = 0.450 between them and is connected to a hanging block of mass m2

= 1.80 kg as shown. When the blocks are released, they move a distance d = 0.50 m, and then m2 hits the floor. Find the speed of the blocks just before m2 hits.

1 2 ; 0i iU m gh m gd K 1 12 2

1 2 1 22 2(0); ;f f nc ncU m gh m g K m v m v E W

i fE E1nc k kW f d m gd

2 21 2 1 1 2 1

1 1

2 2 km gh m gd m gh m v m v m gd

22 1 2 1

1( )

2 km gd m m v m gd

218.82 (4.2 ) 5.29

2J kg v J

8.82 5.291.30 /

2.1

J Jv m s

kg

Potential Energy Curvesand Equipotentials

Potential Energy Curvesand Equipotentials

The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

Potential Energy CurvesPotential Energy Curves

The potential energy curve for a mass and spring

Total mechanical energy