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Hong Kong Polytechnic University
Kinetic Energy and Works
Kinetic Energy (动能):
For a particle of mass m:
K
Work (功):
1 2
mv
2
Work W is the energy transferred to or from an object via a force acting on the
object. For a infinitesimal movement, the work done on the particle is
dW  F  dl  F cosdl
total work:
b
b
a
a
Wab   F  dl   F cosdl
Work done by constant force: W  F  d  Fd cos
Work done by multiple forces: W 
 F d
Work done by spring force: Ws    kxdx 
f
i
1 2 1 2
kxi  kx f
2
2
Hooke’s law: F  kx ; where k is the spring constant.
Power (功率):
Average power: P  W
t
; Instantaneous power: P 
dW F  dl

 F  v  Fv cos
dt
dt
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Kinetic Energy and Works
Example:
A runaway crate slides over a floor toward you. To slow the crate you push against it with a
force F=(2.0N)i+(-6.0N)j while running backward. During your pushing, the crate goes
through a displacement d=(-3.0m)i. How much work has your force done ?
F
W  F  d  -6.0 J
d
Example:
An initially stationary 15.0 kg crate is pulled, via a cable, a distance L=5.70 m up a
frictionless ramp, to a height h of 2.50 m, where it stops. (a) How much work is done on the
crate by its weight mg during the lift? (b) How much work is done on the crate by the force
T applied by the cable, which pulls the crate up the ramp?


(a) : Wg  mgL cos   90  mgL sin   mgh  368 J
(b) : Wg  WT  K  0
WT  Wg  368 J
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Kinetic Energy and Works
Example:
A 500 kg elevator cab is descending with speed vi=4.0 m/s when the cable that controls it
begins to slip, allowing it to fall with constant acceleration a=g/5. (a) During its fall through
a distance d=12 m, what is the work W1 done on the cab by its weight mg? (b) During the 12
m fall, what is the work W2 done on the cab by the upward pull T exerted by the elevator
cable? (c) What is the total work W done on the cab in the 12 m fall? (d) What is the cab’s
kinetic energy at the end of the 12 m fall?
(a) : W1  mgd  5.88 104 J
(b) : mg  T  ma
T  3920 N
W2  Td cos180  4.70 104 J
(c) : W  W1  W2  1.2 104 J
(d ) : W  K  K f  K i
Example:
1
K f  W  K i  W  mvi2  1.60 10 4 J
2
A block whose mass m is 7.5 kg slides on a horizontal frictionless tabletop with a constant
speed v of 1.2 m/s. It is brought momentarily to rest as it compresses a spring in its path. By
what distance d is the spring compressed? The spring constant k is 1500 N/m.
Ws  K  K f  K i
1
1
 kd 2   mv2
2
2
d v
m
 7.4 cm
k
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Kinetic Energy and Works
Example:
You apply a 4.9 N force F to a block attached to the free end of a spring to keep the spring
stretched from its relaxed length by 12 mm. (a) What is the spring constant of the spring? (b)
What force does the spring exert on the block if you stretch the spring by 17 mm? (c) How
much work does the spring force do on the block as the spring is stretched 17 mm as in (b)?
(d) With the spring initially stretched by 17 mm, you allow the block to return to x=0 (the
spring returns to its relaxed state); you then compress the spring by 12 mm. How much work
does the spring force do on the block during this total displacement of the block?
(a) : F  kx; k  410 N/m
(b) : F  kx  6.9 N
(c) : Ws 
1 2 1 2
1
kxi  kx f  0  kx2f  0.059 J
2
2
2
(d ) : Ws 
1 2 1 2 1
kxi  kx f  k xi2  x 2f  0.030 J
2
2
2


College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Kinetic Energy and Works
Example:
Two forces F1 and F2 acting on a box as the box slides rightward across a frictionless floor.
Force F1 is horizontal, with magnitude 2.0 N; force F2 is angled upward by 60 to the floor
and has magnitude 4.0 N. The speed v of the box at a certain instant is 3.0 m/s. (a) What is
the power due to each force acting on the box at that instant, and what is the net power? Is
the net power changing at that instant? (b) If the magnitude of F2 is, instead, 6.0 N, what
now is the power due to each force acting on the box at the given instant, and what is the net
power? Is the net power changing?
(a) : P1  F1v cos180  6.0 W
P2  F2v cos 60  6.0 W
(b) : P2  F2v cos 60  9.0 W
P1  F1v cos180  6.0 W
Pnet  P1  P2  3.0 W
Kinetic energy increases; v increases; P increases.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Kinetic Energy and Works
Homework:
1.
In the figure, a cord runs around two massless, frictionless pulleys;
a canister with mass m=20 kg hangs from one pulley; and you
exert a force F on the free end of the cord. (a) What must be the
magnitude of F if you are to lift the canister at a constant speed?
(b) To lift the canister by 2.0 cm, how far must you pull the free
end of the cord? During that lift, what is the work done on the
canister by (c) your force (via the cord) and (d) the weight mg of
the canister?
2.
A 0.30 kg mass sliding on a horizontal frictionless surface is
attached to one end of a horizontal spring (with k=500 N/m) whose
other end is fixed. The mass has a kinetic energy of 10 J as it
passes through its equilibrium position (the point at which the
spring force is zero). (a) At what rate is the spring doing work on
the mass as the mass passes through its equilibrium position? (b)
At what rate is the spring doing work on the mass when the spring
is compressed 0.10 m and the mass is moving away from the
equilibrium position?
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
Conservative Forces (保守力):
A force is a conservative force if the net work it does on a particle moving along a
closed path from an initial point and then back to that point is zero. Or, equivalently,
it is conservative if its work on a particle moving between two points does not
depend on the path taken by the particle.
Potential Energy (势能):
Potential energy is associated with the configuration of a system in which a
conservative force acts. When the conservative force does work W on a particle
within the system,
xf
dU  x 
U  W    F x dx
F x   
xi
dx
Gravitational Potential Energy (引力势能): U  mgh
Elastic Potential Energy (弹性势能): U x   kx2
Mechanical Energy (机械能): E  K  U
1
2
If only a conservative force within the system does work, then the mechanical
energy E of the system cannot change.
In an isolated system, energy may be transferred from one type to another, but the
total energy of the system remains constant. This is the principle of conservation of
energy (能量守恒原理).
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
If, instead, the system is not isolated, then an external force can change the total
energy of the system by doing work, W  Etot  K  U  Eint
If a nonconservative applied force F does work on particle that is part of a system
having a potential energy, then the work Wapp done on the system by F is equal to
the change E in the mechanical energy of the system, W  K  U  E
app
If a kinetic frictional force fk does work on an object, the change E in the total
mechanical energy of the object and any system containing it is given by, E   f k d
The mechanical energy lost by this transfer is said to be dissipated by fk.
Example:
(16 J)
A 2.0 kg block slides along a frictionless track from point a to
point b. The block travels along a total distance of 2.0 along
the track, and a net vertical distance of 0.80 m. How much
work is done on the block by its weight during the slide?
Example:
(13 m/s)
A child of mass m is released from rest at the top of a water
slide, at height h=8.5 m above the bottom of the slide.
Assuming that the slide is frictionless because of the water on
it, find the child’s speed at the bottom of the slide.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
Example:
The spring of a spring gun is compressed a distance d of 3.2 cm from its relaxed state, and a
ball of mass m=12 g is put in the barrel. With what speed will the ball leave the barrel when
the gun is fired? The spring constant k is 7.5 N/cm. Assume no friction and a horizontal gun
barrel. Also assume that the ball leaves the spring and the spring stops when the spring
reached its relaxed length.
U i  Ki  U f  K f
1 2 1 2
kd  mv
2
2
vd
k
 8.0 m/s
m
Example:
A 61.0 kg bungee-cord jumper is on a bridge 45.0 m above a rive. In its relaxed state, the
elastic bungee cord has length L=25.0 m. Assume that the cord obeys Hooke’s law, with a
spring constant of 160 N/m. (a) If the jumper stops before reaching the water, what is the
height h of her feet above the water at her lowest point? (b) What is the net force on her at her
lowest point?
1
(a) : K  U e  U s  0
d  17.9 m
2
kd 2  mg L  d   0
h  2.1 m
(b) : Fnet  kd  mg  2270 N
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
Example:
A steel block of mass m=40 kg being dragged by a cable up a 30 inclined plane. The applied
force F exerted on the block by the cable has a magnitude of 380 N. The kinetic frictional
force fk acting on the block has a magnitude of 140 N. The block moves through a
displacement d of magnitude 0.50 m along the inclined plane. (a) How much of the
mechanical energy of the block-Earth system is dissipated by the kinetic frictional force fk
during displacement d? (b) What is the work Wg done on the block by its weight during the
displacement? (c) What is the work Wapp done by the applied force F?
(a) : E   f k d  70 J
(b) : Wg  mgh  mgd sin 30  98 J
(c) : Wapp  Fd cos 0  190 J
Concept:
The figure shows three plums that are launched from
the same level with the same speed. One moves
straight upward, one is launched at a small angle to the
vertical, and one is launched along a frictionless
incline. Rank the plums according to their speed when
they reach the level of the dashed line, greatest first.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
Example:
A steel ball whose mass m is 5.2 g is fired vertically downward from a height h1 of 18 m
with an initial speed v0 of 14 m/s. It buries itself in sand to a depth h2 of 21 cm. (a) What is
the change in the mechanical energy of the ball? (b) What is the change in the internal
energy of the ball-Earth-sand system? (c) What is the magnitude of the average force F
exerted on the ball by the sand?
1


(a) : E  K  U   0  mv02   mg h1  h2   1.4 J
2


(b) : Etot  E  Eint  0
(c) :  Fh2  E;
Eint  E  1.4 J
F  6.8 N
Homework:
1.
The figure shows a thin rod, of length L and negligible mass, that can
pivot about one end to rotate in a vertical circle. A heavy ball of mass m is
attached to the other end. The rod is pulled aside through an angle  and
released. As the ball descends to its lowest point , (a) how much work
does its weight do on it and (b) what is the change in the gravitational
potential energy of the ball-Earth system? (c) If the gravitational potential
energy is taken to be zero at the lowest point, what is its value just as the
ball is released?
College Physics----by Dr.H.Huang, Department of Applied Physics

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Hong Kong Polytechnic University Potential Energy & Conservation of Energy
2.
The string in the figure is L=120 cm long, and the distance d to the
fixed peg at point P is 75.0 cm. When the initially stationary ball is
released with the string horizontal as shown, it will swing along the
dashed arc. What is the speed when it reaches (a) its lowest point
and (b) its highest point after the string catches on the peg? (c) If
the ball is to swing completely around the fixed peg, prove that
d>3L/5.
3.
A boy is seated on the top of a hemispherical mound of ice. He is
given a very small push and starts sliding down the ice. Show that
he leaves the ice at a point whose height is 2R/3 if the ice is
frictionless. (Hint: The normal force vanished as he leaves the ice.)
4.
A 4.0 kg bundle starts up a 30 incline with 128 J of kinetic energy.
How far will it slide up the plane if the coefficient of friction is
0.30?
5.
A stone with weight w is thrown vertically upward into the air from
ground level with initial speed v0. If a constant force f due to air
drag acts on the stone throughout its flight, (a) calculate the
maximum height that can be reached by the stone. (b) Calculate
the speed of the stone just before impact with the ground.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Collisions
Collisions (碰撞)
A collision is an isolated event in which two or more bodies (the colliding bodies)
exert relatively strong forces on each other for a relatively short time. These forces
are internal to the colliding-body system and are significantly larger than any external
force during the collision. The laws of conservation of energy and linear momentum
apply, immediately before and after a collision.
Elastic Collision (弹性碰撞)
The kinetic energy of a system of two colliding bodies is conserved.
Inelastic Collision (非弹性碰撞)
The kinetic energy of a system of two colliding bodies is not conserved. For
completely inelastic collision (完全非弹性碰撞), the colliding bodies stick together
and the reduction in kinetic energy is maximum.
Motion of the Center of Mass
The center of mass of a closed, isolated system of colliding bodies is unaffected by
the collision, whether the collision is elastic or inelastic.
Impulse (冲量) -linear momentum theorem:
t
p f  p i  p  J   Ft dt
f
ti
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Collisions
Example:
A pitched 140 g baseball, in horizontal flight with a speed vi of 39 m/s, is struck by a batter.
After leaving the bat, the ball travels in the opposite direction with speed vf, also 39 m/s. (a)
What impulse J acts on the ball while it is in contact with the bat? (b) The impact time t for
the baseball-bat collision is 1.2 ms. What average force acts on the baseball? (c) What is the
average acceleration a of the baseball?
(b) : F  J t  9100 N
(a) : J  p f  pi  mv f  mvi  10.9 kgm/s
(c) : a  F m  6.5 10 4 m/s 2
Example:
As in the above example, the baseball approaches the bat
horizontally at a speed vi of 39 m/s, but now the ball leaves the
bat with a speed vf of 45 m/s at an upward angle of 30 from the
horizontal. What is the average force F exerted on the ball if the
collision lasts 1.2 ms?
J x  p fx  pix  mv fx  vix   10.92 kgm/s
J y  p fy  piy  mv fy  viy   3.150 kgm/s
F  J t  9475 N
tan   J y J x
J  J x2  J y2  11.37 kgm/s
  16
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Example:
Collisions
((a)-0.54m/s; (b)1.5cm; (c)0.72m/s; (d)2.6cm)
Two metal spheres, suspended by vertical cords, initially
just touch. Sphere 1, with m1=30 g, is pulled to the left
to height h1=8.0 cm, and then released. After swinging
down, it undergoes an elastic collision with sphere 2,
whose mass m2=75 g. (a) What is the velocity v1f of
sphere 1 just after the collision? (b) To what height h1
does sphere 1 swing to the left after the collision? (c) What is the velocity v2f of sphere 2 just
after the collision? (d) To what height h2 does sphere 2 swing after the collision?
1
(a) :  mv12i  m1 gh1
 v1i  2 gh1  1.252 m/s
2
m1  m2
2m2
v

v

v2 i
1f
1i
m1v1i  m2v2i  m1v1 f  m2v2 f
m1  m2
m1  m2
1
1
1
1
2
2
2
2m1
m2  m1
m1v1i  m2v2i  m1v1 f  m2v22 f
v

v

v2 i
2f
1i
2
2
2
2
m1  m2
m1  m2
v1 f  0.537 m/s
1
(b) :  m1 gh1  m1v12f
h1  0.0147 m
(c) : v2 f  0.715 m/s
2
1
(d ) :  m2 gh2  m2 v22 f
 h2  0.0261 m
2
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Collisions
Example:
The ballistic pendulum was used to measure the speeds of
bullets before electronic timing devices were developed. The
pendulum consists of a large block of wood of mass M=5.4
kg, hanging from two long cords. A bullet of mass m=9.5 g is
fired into the block, coming quickly to reset. The
block+bullet then swing upward, their center of mass rising a
vertical distance h=6.3 cm before the pendulum comes
momentarily to rest at the end of its arc. (a) What was the
speed v of the bullet just prior to the collision? (b) What is the
initial kinetic energy of the bullet? How much of this energy
remains as mechanical energy of the swinging pendulum?
(a) : mv  M  m V
1
M  mV 2  M  mgh
2
M m
2 gh  630 m/s
m
1
(b) : K b  mv2  1900 J
E  M  m gh  3.3 J
2
v
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Collisions
Example:
Two particles of equal masses have an elastic collision,
the target particle being initially at rest . Show that
(unless the collision is head-on) the two particle will
always move off perpendicular to each other after the
collision.
v1i  v1 f  v 2 f
v12i  v12f  v22 f
Concept:
Drop, in succession, a baseball and a basketball from
about shoulder height above a hard floor, and note how
high each rebounds. Then align the baseball above the
basketball (with a small separation as shown in the
figure) and drop them simultaneously. (a) Is the rebound
height of the basketball now higher or lower than before?
(b) Is the rebound height of the baseball less than or
greater than the sum of the individual baseball and
basketball rebound heights?
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Collisions
Example:
Two skaters collide and embrace, in a completely inelastic
collision. That is, they stick together after impact, as suggested
by the figure, where the origin is placed at the point of collision.
Alfred, whose mass mA is 83 kg, is originally moving east with
speed vA=6.2 km/h. Barbara, whose mass mB is 55 kg, is
originally moving north with speed vB=7.8 km/h. (a) What is the
velocity V of the couple after impact? (b) What is the velocity
of the center of mass of the two skaters before and after the
collision? (c) What is the fractional change in the kinetic energy
of the skaters because of the collision?
(a) : mAv A  mA  mB V cos 
tan  
mB v B
 0.834
m Av A
  39.8
mB vB  mA  mB V sin 
mB v B
V
 4.86 km/h
(b) : V  4.86 km/h
M sin 
1
1
1
(c) : K i  m Av A2  mB vB2  3270 kg  km 2 /h 2
K f  MV 2  1630 kg  km 2 /h 2
2
2
2
K f  Ki
fraction 
 0.50
Ki
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Collisions
Homework:
1.
The last stage of a rocket is traveling at a speed of 7600 m/s. This last stage is made up of
two parts that are clamped together, namely, a rocket case with a mass of 290.0 kg and a
payload capsule with a mass of 150.0 kg. When the clamp is released, a compressed spring
causes the two parts to separate with a relative speed of 910.0 m/s. (a) What are the speeds
of the two parts after they have separated? Assume that all velocities are along the same line.
(b) Find the total kinetic energy of the two parts before and after they separate; account for
any difference.
2.
You are on an iceboat on frictionless, flat ice; you and the boat have a combined mass M.
Along with you are two stones with masses m1 and m2 such that M=6.00m1=12.0m2. To get
the boat moving, you throw the stones rearward, either in succession or together, but in each
case with a certain speed vrel relative to the boat. What is the resulting speed of the boat if
you throw the stones (a) simultaneously, (b) m1 and then m2, and (c) m2 and then m1?
3.
Two long barges are moving in the same direction in still water, one with a speed of 10 km/h
and the other with a speed of 20 km/h. While they are passing each other, coal is shoveled
from the slower to the faster one at a rate of 1000 kg/min. How much additional force must
be provided by the driving engines of each barge if neither is to change speed? Assume that
the shoveling is always perfectly sideways and that the frictional forces between the barges
and the water do not depend on the weight of the barges.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Fluids
State of Matter:
Matter usually has three states – solid, liquid, and gas. Solids have a definite shape
and are not easily deformed; liquids and gases (together called fluids, because they
flow) do not have a definite shape, and take on the shape of their containers. Liquids
are usually incompressible – they will change their shape but not their volume – but
gases are compressible, and their volume will change with pressure.
Pressure:
Even if the bulk fluid is not flowing, the molecules in it are moving. When they collide
with the walls of the container, they exert a force on it; the aggregate of all these
molecular forces results on a net outward force on the walls. This force is always
perpendicular to the surface; if there were a force on the fluid parallel to the surface,
it would flow, and we are assuming that the fluid is static.
The pressure is defined as the force divided by the area over which the force is
exerted; although the force is a vector, the pressure is a scalar. The units of pressure
(N/m2) are pascals (Pa).
Atmospheric pressure: The Earth’s atmosphere is a fluid, and exerts pressure on
all surfaces that are in contact with it. We do not experience this pressure directly, as
the fluids within our bodies are at about the same pressure as the atmosphere, so
there is no net force on us.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Fluids
Pascal’s Principle:
The pressure everywhere in a static fluid must be the same (otherwise there would
be unbalanced forces), as long as the weight of the fluid can be ignored. A change
in pressure in a confined fluid is transmitted everywhere throughout the fluid
(this is true even when you take the weight of the fluid into account); this allows such
things as hydraulic lifts, as a small force exerted on a small area will result in a large
force on a larger area. Of course, since the liquid is incompressible, the small force
will act over a much larger distance than the large force does.
F1 F2

A1 A2
A1d1  A2 d 2
F1d1  F2 d 2
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Fluids
Effect of Gravity:
The density of a substance is defined as:

m
V
For a liquid whose density is constant, the force at a particular depth is increased by
the weight of the fluid above it compared to the pressure at the surface. Taking this
into account gives us the variation of pressure with depth in a fluid.
m  V  Ad
mg  Adg
F
y
 P2 A  P1 A  mg  0
P2  P1  gd
Pressure at a depth d below the surface of
a liquid open to the atmosphere:
P  Patm  gd
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Fluids
Archimedes’ Principle:
A fluid exerts an upward buoyant force on a
submerged object equal in magnitude to the
weight of the volume of fluid displaced by the
object.
FB  F1  F2
FB  P2 A  P1 A
FB  gdA  gV
The Archimedes’ principle is true for objects of any shape, as well as for objects
which are only partly submerged. The net force on the object is then the vector sum
of this upward force and the downward force of gravity due to the object’s own
weight. If this net force is upward, the object will float; if it is downward, it will sink.
This works in air as well as in water, and is what keeps hot-air balloons afloat.
Specific gravity is the ratio of a material’s density to the density of water at 4°C.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Fluids
Homework:
1.
A small statue is painted in black and has a weight of 24.1 N. The owner of the statue
claims it is made of solid gold. When the statue is completely submerged in a container
brimful of water, the weight of the water that spills over the top and into a bucket is
1.25N. Find the density and specific gravity of the statue. Is it a solid gold?
2.
What percentage of a floating iceberg’s volume is above water? The specific gravity of
ice is 0.917 and the specific gravity of the surrounding seawater is 1.025?
3.
A piece of metal is released under water. The volume of the metal is 50.0 cm3 and its
specific gravity is 5.0. What is its initial acceleration?
4.
A fish uses a swim bladder to change its density so it is equal to that of water, enabling it
to remain suspended under water. If a fish has an average density of 1080 kg/m3 and
mass 10.0 g with the bladder completely deflated, to what volume must the fish inflate
the swim bladder in order to remain suspended in seawater of density 1060 kg/m3?
College Physics----by Dr.H.Huang, Department of Applied Physics
24
Hong Kong Polytechnic University
Waves I
Waves:
Wave is characterized as some sort of disturbance that travels away from its
source. It has a broad distribution of energy, filling the space through which it
passes. In other words, energy is transmitted when the wave is propagating.
Types of Wave:
Mechanical Waves: water waves, sound waves and seismic waves, …
Electromagnetic Waves: light, radio and television wave, microwaves and x-rays,…
Matter Waves: electrons, protons and other fundamental particles travel as
waves…
Intensity:
Average power per unit area carried by the wave past a surface perpendicular to
the wave’s direction of propagation. Its unit is W/m2.
If a point source emits uniformly in all directions, assuming no reflection or
absorption, the intensity at a distance r to the source follows the inverse square law,
I
P
4r 2
where P is the emitted power of the source.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Transverse Waves: particles oscillate
perpendicular to the direction of propagation
Longitudinal Waves: particles oscillate parallel
to the direction of propagation
Transverse Wave Speed on Stretched String: v  

where  is the string tension and  is the linear density (m/L)
Velocity of Longitudinal Waves: v 
E

where E is the elastic modulus and ρ is the density.
Sound Waves: v  B 
where B is the bulk modulus and  is the density.
In air at 20C, the speed of sound is 343 m/s.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
y
ym
ym
yx, t   ym sin kx  t 
Sinusoidal Waves:
amplitude
2
wave length:  
;
k
wave speed: v 

k


T
angular wave
number
period: T 
2

;
phase
angular
frequency
frequency: f 
1 

T 2
 f
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Traveling Wave:
Waves I
yx, t   f kx  t 
Example:
A sinusoidal wave traveling along a string described by yx, t   0.00327 sin 72.1x  2.72t 
in which the numerical constants are in SI units (0.00327 m, 72.1 rad/m, and 2.72 rad/s). (a)
What is the amplitude of this wave? (b) What are the wavelength, period, and frequency of
this wave? (c) What is the speed of this wave? (d) What is the displacement y at x=22.5 cm
and t=18.9 s? What is the transverse speed and the transverse acceleration at that position and
at that time?
(a) : ym  3.27 mm

(c ) : v 
u
2
 8.71 cm;
k

k
 3.77 cm/s
(b) : k  72.1 rad/m and   2.72 rad/s
T
2

 2.31 s;
f 
1
 0.433 Hz
T
(d ) : y  1.92 mm
y
u
 ym coskx  t   7.20 mm/s ; a y 
  2 ym sin kx  t   14.2 mm/s 2
t
t
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Average Power:
The average power, or average rate at which energy is transmitted by a sinusoidal
wave on a stretched string is,
where ym is amplitude.
1
2 2
P
2
v ym
Concept:
The figure above shows two situations in which the same string is put under tension by a
suspended mass of 5 kg. In which situation will the speed of waves sent along the string be
greater?
Example:
A string has a linear density  of 525 g/m and is stretched with a tension  of 45 N. A wave
whose frequency f and amplitude ym are 120 Hz and 8.5 mm, respectively, is traveling along
the string. At what average rate is the wave transporting energy along the string?
  2f  754 rad/s
P
v     9.26 m/s
1
v 2 ym2  100 W
2
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Superposition of Waves (波的叠加):
Overlapping waves algebraically add to produce a resultant wave. Overlapping
waves do not in any way alter the travel of each other. yx, t   y1 x, t   y2 x, t 
Interference (干涉) of Waves:
Two waves: y1 x, t   ym1 sin kx  t  1 
Resultant:
y2 x, t   ym 2 sin kx  t  2 
yx, t   ym sin kx  t   
ym2  ym2 1  ym2 2  2 ym1 ym 2 cos2  1 
tan  
ym1 sin 1  ym 2 sin 2
ym1 cos 1  ym 2 cos 2
When 2  1  0 , the two waves are in-phase, the resultant wave has the maximum
amplitude. This type of interference is called fully constructive interference.
When 2  1   , the two waves are anti-phase, the resultant wave has the minimum
amplitude. This type of interference is called fully destructive interference.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Example:
Two identical waves, moving in the same direction along a stretched string, interfere with
each other. The amplitude of each wave is 9.8 mm. (a) If the phase difference  between
them is 100, what is the amplitude ym of the resultant wave due to the interference of these
two waves? (b) What phase difference, in radians and in wavelengths, will give the resultant
wave an amplitude of 4.9 mm?
(a ) : 2  1  100
ym1  ym 2  9.8
ym  13 mm
(b) : If ym  4.9 then 2  1  2.6 rad
In wavelength the difference is
2.6
 0.4 wavelength
2
Example:
Two waves y1(x,t) and y2(x,t) have the same wavelength and travel in the same direction
along a string. Their amplitudes are ym1=4.0 mm and ym2=3.0 mm, and their phase constants
are 0 and /3 rad, respectively. What are the amplitude and phase constant  of the resultant
wave?
ym  6.1 mm
  0.44 rad
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Standing Wave (驻波):
Consider two waves: y1 x, t   ym sin kx  t 
The resultant wave is:
y2 x, t   ym sin kx  t 
yx, t   2 ym sin kx cos t
This wave is not traveling and the
amplitude varies with position.
Nodes at x  n
Waves I
2ym

2
1

Antinodes at x   n  
2 2

-2ym
Reflection at a Boundary:
For a clamped end: phase shift of 
For a free end: no phase shift
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Resonance:
For a stretched string between two clamps separated by a
distance L, at certain resonant frequencies, standing waves
can be formed.
wavelength:  
2L
n
Resonant frequency: f 
nv
2L
The oscillation with the lowest frequency (n=1) is called the
fundamental mode or the first harmonic. The second
harmonic is the oscillation mode with n=2; and so on. n is
called the harmonic number of the nth harmonic and the
collection of the frequencies associated with these modes is
called the harmonic series.
Concept:
Two strings of equal length but unequal linear
densities are tied together with a knot and
stretched between two supports. A particular frequency happens to produce a standing wave
on each length, with a node at the knot, as shown in the figure. Which string has the greater
liner density?
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves I
Example:
As shown in the figure, a string tied to a sinusoidal vibrator at P and running over a support
at Q, is stretched by a block of mass m. The separation L between P and Q is 1.2 m, the linear
density of the string is 1.6 g/m, and the frequency f of the vibrator is fixed at 120 Hz. The
amplitude of the motion at P is small enough for that point to be considered a node. A node
also exists at Q. (a) What mass m allows the vibrator to set up the fourth harmonic on the
string? (b) What standing wave mode is set up if m=1.00 kg?
(a) : v 

mg



nv
f 
2L
4L2 f 2 
(b) : m 
n2 g
4 L2 f 2 
m
 0.846 kg
2
n g
n  3.7
It’s impossible to set up a standing wave in the string.
College Physics----by Dr.H.Huang, Department of Applied Physics
34
Hong Kong Polytechnic University
Waves I
Homework:
1.
2.
3.
A sinusoidal transverse wave is traveling along a string
toward decreasing x. The figure shows a plot of the
displacement as a function of position at time t=0. The string
tension is 3.6 N, and its linear density is 25 g/m. Find (a) the
amplitude, (b) the wavelength, (c) the wave speed, and (d) the
period of the wave. (e) Find the maximum speed of a particle
in the string. (f) Write an equation describing the traveling
wave.
In the figure a, string 1 has a linear density of 3.00 g/m, and
string 2 has a linear density of 5.00 g/m. They are under
tension owing to the hanging block of mass M=500 g. (a)
Calculate the wave speed in each string. (b) The block is now
divided into two blocks (with M1+M2=M) and the apparatus
rearranged as shown in figure b. Find M1 and M2 such that the
wave speeds in the two strings are equal.
a
b
Vibration from a 600 Hz tuning fork sets up standing waves in a string clamped at both
ends. The wave speed for the string is 400 m/s. The standing wave has four loops and
an amplitude of 2.0 mm. (a) What is the length of the string? (b) Write an equation for
the displacement of the string as a function of position and time.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves II
Beats:
The superposition of sound waves s1  sm cos 1t
and s2  sm cos 2t with slight difference in frequency
is, s  2sm cos t cos t
where
1
1
 
2
1  2 

2
1  2 
Beat frequency: beat  1  2
or, f beat  f1  f 2
Doppler Effect:
f f
v  vD
v  vS
vD is the speed of detector relative to the medium.
vS is the speed of source relative to the medium.
If the source speed relative to the medium exceeds
the speed of sound in the medium, the Doppler
equation no longer applies. Shock waves result. The
half angle of the wavefront is given by,
v
sin  
vS
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves II
Example:
A toy rocket moves at a speed of 242 m/s directly toward a stationary pole (through
stationary air) while emitting sound waves at frequency f=1250 Hz. (a) What frequency f is
sensed by a detector that is attached to the pole? (b) Some of the sound reaching the pole
reflects back to the rocket, which has an onboard detector. What frequency f does it detect?
(a) :
f f
v
 4250 Hz
v  vS
(b) :
f   f 
v  vD
 7240 Hz
v
Example:
Suppose a bat flies toward a moth at speed vb=9.0 m/s, while the moth flies toward the bat
with speed vm=8.0 m/s. The bat emits ultrasonic waves of frequency fbe that reflect from the
moth back to the bat with frequency fbd. The bat adjusts the emitted frequency fbe until the
returned frequency fbd is 83 kHz, at which the bat’s hearing is best. (a) What is the frequency
fm of the waves heard and reflected by the moth? (b) What is the frequency fbe emitted by the
bat?
(a) : moth being the source
(b) : bat being the source
f bd  f m
f m  f be
v  vb
v  vm
v  vm
v  vb
 f m  f bd
 f be  f m
v  vm
 79 Hz
v  vb
v  vb
 75 Hz
v  vm
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Doppler Effect for Light:
Waves II
f   f 1  u c
u is the relative speed between a light source and a detector.
Proof:
1
c  vD
 v  v 
 v  v
f f
 f 1  D 1  S   f 1  D 1  S
c  vS
c 
c
c 
c


 v v 
 u
 f 1  D S   f 1  
c 
 c




v v 
 v v
f 1  D S  D 2 S 
c
c 

The relative speed u is related to the Doppler shift in wavelength  by u 


c
If the source and detector are approaching each other, there is a blue shift in
wavelength (frequency increases). If the source and detector are moving away
from each other, there is a red shift.
College Physics----by Dr.H.Huang, Department of Applied Physics
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Hong Kong Polytechnic University
Waves II
Homework:
1.
2.
3.
An experimenter wishes to measure the speed of sound in an aluminum rod 10 cm long
by measuring the time it takes for a sound pulse to travel the length of the rod. If results
good to four significant figures are desired, how precisely must the length of the rod be
known and how closely must she be able to resolve time intervals? (The speed of sound
in aluminum is 6420 m/s).
Two identical piano wires have a fundamental frequency of 600 Hz when kept under the
same tension. What fractional increase in the tension of one wire will lead to the
occurrence of 6 beats/s when both wires oscillate simultaneously?
Two identical tuning forks can oscillate at 440 Hz. A person is located somewhere on
the line between them. Calculate the beat frequency as measured by this individual if (a)
she is standing still and the tuning forks both move to the right at 30.0 m/s, and (b) the
tuning forks are stationary and the listener moves to the right at 30.0 m/s.
College Physics----by Dr.H.Huang, Department of Applied Physics
39