Chapters 6, 7 Energy
description
Transcript of Chapters 6, 7 Energy
Chapters 6, 7
Energy
Energy
• What is energy?
• Energy - is a fundamental, basic notion in physics
• Energy is a scalar, describing state of an object or a system
• Description of a system in ‘energy language’ is equivalent to a description in ‘force language’
• Energy approach is more general and more effective than the force approach
• Equations of motion of an object (system) can be derived from the energy equations
Scalar product of two vectors
• The result of the scalar (dot) multiplication of two vectors is a scalar
• Scalar products of unit vectors
cosABBA
ii ˆˆ 1ˆˆ jj
ji ˆˆ
1ˆˆ kk
0ˆˆ ki 0ˆˆ kj
0cos11 1
90cos11 0
Scalar product of two vectors
• The result of the scalar (dot) multiplication of two vectors is a scalar
• Scalar product via unit vectors
)ˆˆˆ)(ˆˆˆ( kBjBiBkAjAiABA zyxzyx
zzyyxx BABABABA
cosABBA
Some calculus
• In 1D case
dt
dxv
dt
dva
dt
dx
dx
dv
dx
vdv
dx
vd
2
2
dx
mvd
Fma net
2
2
2
2mvddxFnet
Some calculus
• In 1D case
• In 3D case, similar derivations yield
• K – kinetic energy
2
2mvddxFnet
2
2mvdrdFnet
Kd
2
2mvK
Kinetic energy
• K = mv2/2
• SI unit: kg*m2/s2 = J (Joule)
• Kinetic energy describes object’s ‘state of motion’
• Kinetic energy is a scalar
James Prescott Joule(1818 - 1889)
Work-kinetic energy theorem
• Wnet – work (net)
• Work is a scalar
• Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy
• Work is done on the object by a force
dKmv
drdFnet
2
2
f
i
r
r netif rdFKK
netW
Work: graphical representation
• 1D case: Graphically - work is the area under the curve Fx(x)
f
i
x
x xdxFW
f
i
x
xx
xxF
0lim
Chapter 6Problem 52
A force with magnitude F = a√x acts in the x-direction, where a = 9.5 N/m1/2. Calculate the work this force does as it acts on an object moving from (a) x = 0 to x = 3.0 m; (b) 3.0 m to 6.0 m; and (c) 6.0 m to 9.0 m.
Net work vs. net force
• We can consider a system, with several forces acting on it
• Each force acting on the system, considered separately, produces its own work
• Since
f
i
r
r kk rdFW
) ( sumvectorFFk
knet
) ( sumscalarWWk
knet
f
i
r
r netnet rdFW
f
i
r
rk
k rdF
k
r
r k
f
i
rdF
Work done by a constant force
• If a force is constant
• If the displacement and the constant force are not parallel
f
i
r
rrdFW
f
i
r
rrdF
rF
coscos FdrFrFW
Work done by a constant force
coscos FdrFrFW
Work done by a spring force
• Hooke’s law in 1D
• From the definition of work
kxFs
f
i
x
x ss dxFW f
i
x
xkxdx
22
22fi
kxkx
Work done by the gravitational force
• Gravity force is ~ constant near the surface of the Earth
• If the displacement is vertically up
• In this case the gravity force does a negative work (against the direction of motion)
cosmgdWg
180cosmgdWgmgd
Lifting an object
• We apply a force F to lift an object
• Force F does a positive work Wa
• The net work done
• If in the initial and final states the object is at rest, then the net work done is zero, and the work done by the force F is
mgdWW ga
gaifnet WWKKKW
Power
• Average power
• Instantaneous power – the rate of doing work
• SI unit: J/s = kg*m2/s3 = W (Watt)
t
WPavg
dt
dWP
James Watt(1736-1819)
dt
dWP
dt
rdF
vF
cosFvP
Chapter 6Problem 36
A 75-kg long-jumper takes 3.1 s to reach a prejump speed of 10 m/s. What’s his power output?
Conservative forces
• The net work done by a conservative force on a particle moving around any closed path is zero
• The net work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle
02,1, baab WW
2,1, baab WW
2,2, baab WW
2,1, abab WW
Conservative forces: examples
• Gravity force
• Spring force
0 downup mghmgh
022
22
leftright kxkx
Potential energy
• For conservative forces we introduce a definition of potential energy U
• The change in potential energy of an object is being defined as being equal to the negative of the work done by conservative forces on the object
• Potential energy is associated with the arrangement of the system subject to conservative forces
WU
Potential energy
• For 1D case
• A conservative force is associated with a potential energy
• There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force
• In 3D
if UUU
CdxxFxU )()(dx
xdUxF
)()(
kz
zyxUj
y
zyxUi
x
zyxUzyxF ˆ),,(ˆ),,(ˆ),,(),,(
f
i
x
xdxxFW )(
Gravitational potential energy
• For an upward direction the y axis
f
i
y
ydymgyU )()( ymgmgymgy if
mgyyU g )(
Gravitational potential energy
mgyyU g )(
Elastic potential energy
• For a spring obeying the Hooke’s law
f
i
x
xdxkxxU )()(
22
22
if kxkx
2)(
2kxxU s
Chapter 7Problem 37
A particle moves along the x-axis under the influence of a force F = ax2 + b, where a and b are constants. Find its potential energy as a function of position, taking U = 0 at x = 0.
Conservation of mechanical energy
• Mechanical energy of an object is
• When a conservative force does work on the object
• In an isolated system, where only conservative forces cause energy changes, the kinetic and potential energies can change, but the mechanical energy cannot change
UKEmec
WK WU UK )( ifif UUKK
iiff UKUK imecfmec EE ,,
Conservation of mechanical energy
• From the work-kinetic energy theorem
• When both conservative a nonconservative forces do work on the object
netWK
nccnet WWW ncWU K
ncWUK
Internal energy
• The energy associated with an object’s temperature is called its internal energy, Eint
• In this example, the friction does work and increases the internal energy of the surface
Chapter 7Problem 53
A spring of constant k = 340 N/m is used to launch a 1.5-kg block along a horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18 cm, how far does the block slide?
Conservation of mechanical
energy: pendulum
Potential energy curve
dx
xdUxF
)()(
Potential energy curve: equilibrium points
Unstable equilibrium
Neutral equilibrium
Stable equilibrium
Questions?
Answers to the even-numbered problems
Chapter 6
Problem 14:
9.6 × 106 J
Answers to the even-numbered problems
Chapter 6
Problem 40:
The hair dryer consumes more energy.
Answers to the even-numbered problems
Chapter 6
Problem 50:
360 J
Answers to the even-numbered problems
Chapter 7
Problem 14:
(a) 7.0 MJ (b) 1.0 MJ
Answers to the even-numbered problems
Chapter 7
Problem 24:
(a) ± 4.9 m/s (b) ± 7.0 m/s(c) ≈ 11 m
Answers to the even-numbered problems
Chapter 7
Problem 38:
95 m