Walker 4th Edition Ch8 Notes

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Lecture Outline

Chapter 8

Physics, 4th Edition

James S. Walker


Chapter 8

Potential Energy and Conservation of Energy


Units of Chapter 8

•  Conservative and Nonconservative Forces

•  Potential Energy and the Work Done by Conservative Forces

•  Conservation of Mechanical Energy

•  Work Done by Nonconservative Forces

•  Potential Energy Curves and Equipotentials


8-1 Conservative and Nonconservative Forces

Conservative force: the work it does is stored in the form of energy that can be released at a later time

Example of conservative forces: gravity, spring force

Example of a nonconservative force: friction

Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force



Work done by gravity on a closed path is zero:



Work done by friction on a closed path is not zero:



The work done by a conservative force is zero on any closed path:


8-2 The Work Done by Conservative Forces

If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf; in the meantime, we say the energy is stored as potential energy.

(8-1)


8-2 The Work Done by Conservative Forces Gravitational potential energy:

Potential energy is a property of the entire system.


8-2 The Work Done by Conservative Forces Example 8-1: Potential energy of diver with m = 65 kg, y = 3.m . Let U=0 at y=0.

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U = mgy = (65 kg)(9.81m/s2)(3.0 m) =1900 J


8-2 The Work Done by Conservative Forces Example 8-2: Climber with m = 82 kg makes a total ascent of 4301 m . What is the change of potential energy in the last 100m of the ascent?


8-2 The Work Done by Conservative Forces Example 8-2: Climber with m = 82 kg makes a total ascent of 4301 m . What is the change of potential energy in the last 100m of the ascent?

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We choose as U = 0 the point of last ascent and we choose this as the origin.

U = mgyU(y =100 m) = (82 kg)(9.81 m/s2)(100 m) = 80,400 J

You get the same answer if you chose the origin as the beginning of the 4301 m ascent.


8-2 The Work Done by Conservative Forces

Springs: (8-4)

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For Uf = 0 (the equilibrium point of the spring)we can write the the potential energy stored in a spring system is:

U = 12

kx2

This is positive for stretched and compressed springs.


8-3 Conservation of Mechanical Energy Definition of mechanical energy:

(8-6)

Using this definition and considering only conservative forces, we find:

Or equivalently:


8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:


8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:

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Ei = Ef ⇒ Ki +Ui = Kf +Uf ⇒

0 +mgh = 12mv2 + 0⇒

v = 2gh


8-3 Conservation of Mechanical Energy Example 8-5 Graduation Fling. m=0.12 kg, vi = 7.85 m/s. (a) v(y=1.18m)=? (b) Show the Ei=Ef


8-3 Conservation of Mechanical Energy

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vf2 = vi

2 − 2gy = vf = 6.2 m/s

Ei = Ki + Ui =12

mvi2 + 0 = 3.7J

Ef = Kf + Uf =12

mvf2 + mgy = 3.7 J

Example 8-5 Graduation Fling. m=0.12 kg, vi = 7.85 m/s. (a) v(y=1.18m)=? (b) Show the Ei=Ef


8-3 Conservation of Mechanical Energy Example 8-6 Ball of m=0.15 kg with vi=36m/s is caught at h=7.2m. (a) kinetic energy and (b) speed of ball when caught?


8-3 Conservation of Mechanical Energy Example 8-6 Ball of m=0.15 kg with vi=36m/s is caught at h=7.2m. (a) kinetic energy and (b) speed of ball when caught?

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Conservation of mechanical energy since friction is ignored :Ei = Ef ⇒ Ki + Ui = Kf + Uf

12

mvi2 =

12

mvf2 + mgh

(a) Kf =12

mvf2 =

12

mvi2 −mgh

Kf = (0.5)(0.15 kg)(36m/s)2 - (0.15 kg)(9.81 m/s2)(7.2 m)Kf = 97J -11J = 86J

(b) vf =2Kf

m= 34m/s


8-3 Conservation of Mechanical Energy Example 8-7 55 kg skateboarder Tony Hawk enters a ramp with vi=6.5 m/s and exits with vf=6.5 m/s .What is the height of the ramp?


8-3 Conservation of Mechanical Energy Example 8-7 55 kg skateboarder Tony Hawk enters a ramp with vi=6.5 m/s and exits with vf=6.5 m/s .What is the height of the ramp?

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Conservation of mechanical energy since friction is ignored :

Ei = Ef

12

mvi2 =

12

mvf2 + mgh

h = 12g

vi2 − vf

2( ) =1.3 m


8-3 Conservation of Mechanical Energy Example 8-8 Spring Time. m = 1.70 kg, k = 955 N/m, x(v=0)=-4.60 cm. What is the initial speed of the block?


8-3 Conservation of Mechanical Energy Example 8-8 Spring Time. m = 1.70 kg, k = 955 N/m, x(v=0)=-4.60 cm. What is the initial speed of the block?

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Ki +Ui = Kf + Uf

12

mvi2 + 0 = 0 +

12

kd2

vi =kd2

m= d k

m= (4.6 cm) 955N /m

1.70kg=1.09m/s


8-3 Conservation of Mechanical Energy Example : Speed of block. m = 1.70 kg, k = 955 N/m, y(v=0)=-4.60 cm. What is v(y=0) of the block?

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Ki +Ui = Kf + Uf

0 + 0 +12

kd2 =12

mv2 + mgd

12

mv2 =12

kd2 −mgd⇒ v2 =kd2

m− 2gd

v = kd2

m− 2gd = 0.535m/s


8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved:

Solving,

(8-9)


8-4 Work Done by Nonconservative Forces Example 8-9: A leaf falls in the forest. m = 17 g, h = 5.3m, vf=1.3 m/s. What is the work done on the leaf by all the nonconservative forces?


8-4 Work Done by Nonconservative Forces Example 8-9: A leaf falls in the forest. m = 17 g, h = 5.3m, vf=1.3 m/s. What is the work done on by all the nonconservative forces?

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The nonconservative forces will change the total mechanical energy by an amount equal to the work they do on the leaf.Wnc = ΔE = Ef - E i

Next we find the change of the total mechanical energy :

Ef = 12

mvf2 =

12

(17.0 g)(1.3 m/s)2

E i = mgh = (17.0 g)(9.81 m/s2)(5.3 m)Wnc = Ef - E i = 0.014J − 0.884J = −0.870J

We expect the work to be negative since all the nc forces are in a direction opposite to the displacement of the leaf and the total energy of the system is reduced.


8-4 Work Done by Nonconservative Forces Example : Find the Diver’s Depth m=95 kg, h=3.0m. At what depth will the diver come to rest if the work done by the nonconservative forces is Wnc = -5120 J

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Wnc = ΔE = Ef - E i

Ef = 0Ei = mg(h +d) = mgh +mgdWnc = Ef - E i = −mgh −mgd⇒ mgd = −Wnc −mgh

d = −Wnc

mg− h = 5120 J

(95.0 kg)(9.81 m/s2)− 3.0m = 2.49 m


8-4 Work Done by Nonconservative Forces Example 8-10:Landing with a thud. m1 = 2.40 kg, m2=1.80 kg, d=0.500 m, µk=0.45. What is the speed of the blocks just before m2 lands.


8-4 Work Done by Nonconservative Forces Example 8-10:Landing with a thud. m1 = 2.40 kg, m2=1.80 kg, d=0.500 m, µk=0.45. What is the speed of the blocks just before m2 lands.

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Wnc = ΔE = Ef - E i

Wnc = - Fkd = -µkNd = -µkm1gd

E = U +K (for both objects)E i = m1gh +m2gd +0 +0 (initial mechanical energy)

Ef = m1gh +0 + 12

m1v2 +

12

m2v2(final mechanical energy)

ΔE = Ef - Ei =12

m1v2 +

12

m2v2 −m2gd

ΔE = Wnc ⇒12

m1v2 +

12

m2v2 −m2gd = -µkm1gd⇒

v =2gd m2 - µkm1( )

m1 +m2( )=1.30m/s


8-5 Potential Energy Curves and Equipotentials

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



The potential energy curve for a spring:



Contour maps are also a form of potential energy curve:


Summary of Chapter 8

•  Conservative forces conserve mechanical energy

•  Nonconservative forces convert mechanical energy into other forms

•  Conservative force does zero work on any closed path

•  Work done by a conservative force is independent of path

•  Conservative forces: gravity, spring


Summary of Chapter 8 •  Work done by nonconservative force on closed path is not zero, and depends on the path

•  Nonconservative forces: friction, air resistance, tension

•  Energy in the form of potential energy can be converted to kinetic or other forms

•  Work done by a conservative force is the negative of the change in the potential energy

•  Gravity: U = mgy

•  Spring: U = ½ kx2


Summary of Chapter 8

•  Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces

•  Nonconservative forces change a system’s mechanical energy

•  Work done by nonconservative forces equals change in a system’s mechanical energy

•  Potential energy curve: U vs. position

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Walker 4th Edition Ch8 Notes