Chapter 11
Elasticity And Periodic Motion
Stress characterizes the strength of the force
associated with the stretch, squeeze, or twist,
usually on a “force per unit area” basis.
Strain describes the deformation that occurs.
When the stress and strain are small enough, we
often find that the two are directly proportional.
The general pattern that emerges can be
formulated as
Stress/Strain = Constant.
Experiments have shown that, for a sufficiently
small tensile or compressive stress, stress and
strain are proportional, as stated by Hooke’s law.
The corresponding proportionality constant, called
Young’s modulus (denoted by Y ), is given by
Y = (Tensile stress)/(Tensile strain) or
Y = (Compressive stress)/(Compressive strain),
Or
Y = (F /A) / (Δl / l0)
• Strain is a pure number, so the units of Young’s
modulus are the same as those of stress: force per
unit area.
• Young’s modulus is a property of a specific material,
rather than of any particular object made of that
material.
• A material with a large value of Y is relatively
un-stretchable; a large stress is required by a given
strain.
Modeling an Elastic Material as a Spring
Fperpendicular = (YA Δl) / l0
Let k = (YA) / l0 , Δl = x and Fperpendicular = Fx ;
then we have Fx = kx.
The pressure in a fluid, denoted by p, is the force
Fperpendicular per unit area A transmitted across any
cross section of the fluid, against a wall of its
container, or against a surface of an immersed
object:
p = Fperpendicular/A
When a solid object is immersed in a fluid and both
are at rest, the forces that the fluid exerts on the
surface of the object are always perpendicular
to the surface at each point.
When Hooke’s law is obeyed, the volume strain is
proportional to the volume stress (change in pressure).
The corresponding constant ratio of stress to strain is
called the bulk modulus, denoted by B.
When the pressure on an object changes by a small
amount Δp, from p0 to p0 + Δp, and the resulting volume
strain is ΔV/V, Hooke’s law takes the form
B = - (Δp)/(ΔV/V0)
We include a minus sign in this equation because an
increase in pressure always causes a decrease in
volume. In other words, when Δp is positive, ΔV is
negative.
Shear Stress and Strain
If the forces are small enough so that Hooke’s law
is obeyed, the shear strain is proportional to the
shear stress. The corresponding proportionality
constant (ratio of shear stress to shear strain), is
called the shear modulus, denoted by S:
S = Shear stress / Shear strain
= (Fparallel / A) / (x / h)
= (Fparallel / A) / Φ
Periodic Motion
Fx = -kx
Using Newton’s second law,
max = -kx
OR
ax = -(k/m)x
• Amplitude, A
• Cycle
• Period, T
• Frequency, f
f = 1/T
SI unit : Hertz (Hz) = cycle/s = 1/s
• Angular frequency, ω
ω = 2πf = 2π/T
Energy in Simple Harmonic Motion
Conservation of Mechanical Energy
E = (1/2) mvx2 + (1/2)kx2 = constant
A useful equation
When x = ± A, vx = 0. At this point, the energy is
entirely potential energy and E = (1/2)kA2 .
E = (1/2)kA2 = (1/2) mvx2 + (1/2)kx2
vx = ±
k/m
A2 – x2
We can use this equation to find the magnitude of the
velocity for any given position x.
Equations of Simple Harmonic Motion
Periodic Motion
The relationship
between uniform
circular motion and
simple harmonic
motion.
Position of the Shadow
as a Function of Time
x = A cos θ
x = A cos(ωt)
x = A cos [(2π/T )t]
SI unit: m
Velocity in Simple
Harmonic Motion
v = -Aω sin(ωt)
SI unit: m/s
Maximum speed of
the mass is
vmax = Aω
Period of a Mass on a Spring
From equation ax = -Aω2 cos(ωt)
ax = -Aω2 (maximum acceleration)
Also, we know that
ax = -(kx)/m
ax = -(kA)/m (maximum acceleration)
From equations (1) and (2)
-Aω2 = -kx/m
ω2 = k/m
ω = k/m = 2πf = 2π/T
T = 2π m / k
SI unit: s
(1)
(2)
The Simple
Pendulum
The restoring force F at each point is the component of
force tangent to the circular path at that point:
F = -mgsinθ
If the angle is small, sinθ is very nearly equal to θ (in
radians).
F = -mgθ = -mgx/L
F = -(mg/L)x
The restoring force F is then proportional to the coordinate
x for small displacements, and the constant mg/L
represents the force constant k.
K=mg/L
NOTE that the frequency does NOT depend
on the mass but on the length of the pendulum
A 0.85-kg mass attached to a vertical spring of
force constant 150 N/m oscillates with a
maximum speed of 0.35 m / s. Find the
following quantities related to the motion of the
mass: (a) the period, (b) the amplitude, (c) the
maximum magnitude of the acceleration.
A peg on a turntable moves with a constant linear
speed of 0.67 m / s in a circle of radius 0.45 m. The
peg casts a shadow on a wall. Find the following
quantities related to the motion of the shadow: (a)
the period, (b) the amplitude, (c) the maximum speed,
and the maximum magnitude of acceleration.
Ch. 11, Problem 42.
An object of unknown mass is attached to an ideal
spring with force constant 120 N/m and is found to
vibrate with frequency of 6.00 Hz. Find
(a) period,
(b) the angular frequency, and
(c) the mass of this object.
Ch. 11, Problem 49
(a)If a pendulum has period T and you double its
length, what is its new period in terms of T?
(b)If a pendulum has length L and you want to
triple its frequency, what should be its length
in terms of L?
(c)Suppose a pendulum has a length L and period
T on earth. If you take it to a planet where
the acceleration of freely falling objects is
ten times what it is on earth, what should you
do to the length to keep the period the same
as on earth?
(d) If you do not change the pendulum’s length in
part (c ), what is its period on that planet in terms
of T?
(e) If a pendulum has a period T and you triple the
mass of its bob, what happens to the period (in
terms of T)?
CHAPTER 11, PROBLEM 24
Find the period, frequency, and angular frequency
of (a) the second hand and (b) the minute hand of a
wall clock.
CHAPTER 11, Problem 47
A certain simple pendulum has a period on earth of
1.60 s. What is the period on the surface on Mars,
where the acceleration due to gravity is 3.71 m/s2 ?
Chapter 11, Problem 33
A mass is oscillating with amplitude A at the end
of a spring. How far (in terms of A) is this mass
from equilibrium position of the spring when the
elastic potential energy equals the kinetic energy?
Chapter 11, Problem 34
(a)If a vibrating system has total energy E0, what
will its total energy be (in terms of E0) if you
double the amplitude of vibration?
(b)If you want to triple the total energy of a
vibrating system with amplitude A0, what should
its new amplitude be (in terms of A0)?
Chapter 11, Problem 58
An object suspended from a spring vibrates with
simple harmonic motion. At an instant when the
displacement of the object is equal to one-half the
amplitude, what fraction of the total energy of the
system is kinetic and what fraction is potential?
After you pick up a spare, your bowling ball rolls without
slipping back toward the ball rack with a linear speed of 2.85
m/s. To reach the rack, the ball rolls up a ramp that rises
through a vertical distance of 0.53 m. (a) What is the linear
speed of the ball when it reaches the top of the ramp? (b) If
the radius of the ball were increased, would the speed found
in part (a) increase, decrease, or stay the same? Explain.
CHAPTER 12
MECHANICAL WAVES AND SOUND
A disturbance that propagates from one place
to another is referred to as a wave.
Waves propagate with well-defined speeds
determined by the properties of the material
through which they travel.
Waves carry energy.
In a transverse wave individual particles move
at right angles to the direction of wave
propagation.
In a longitudinal wave individual particles
move in the same direction as the wave
propagation.
A wave on a string
As a wave on a
string moves
horizontally, all
points on the
string vibrate in
the vertical
direction.
Water waves from a disturbance.
The water wave has
characteristics of both
transverse and
longitudinal waves.
Wavelength,
Frequency, and
Speed
Sound produced by a speaker
REFLECTIONS AND SUPERPOSITION
A reflected wave pulse: fixed end
A reflected wave pulse: free end
Sound Waves
Speed of Sound in Air
v = 343 m/s
The frequency of sound determines its
pitch. High-pitched sounds have high
frequencies; low-pitched sounds have
low frequencies.
Human hearing extends from 20 Hz to
20, 000 Hz. Sounds with frequencies
above this range are referred to as
ultrasonic, while those with frequencies
lower than 20 Hz are classified as
infrasonic.
Constructive Interference
Destructive Interference
Figure 14-22
Interference with Two Sources
In phase/opposite phase: Two sources are in phase if they
both emit crests at the same time. Sources have opposite
phase if one emits a crest at the same time other emits a
trough.
Constructive interference occurs when the path length
from the two sources differs by 0, λ, 2λ, 3λ, …….
Destructive interference occurs when the path length from
the two sources differs by λ/2, 3λ/2, 5λ/2, …….
The loudness of a
sound is
determined by its
intensity; that is;
by the amount of
energy that passes
through a given
area in a given
time.
I = E/(At)
Or
I = P/A
I1 = P/(4πr12)
I2 = P/(4πr22)
Chapter 3, Problem 50
A cart carrying a vertical missile
launcher moves horizontally at a
constant velocity of 30.0 m/s to the
right. It launches a rocket
vertically upward. The missile has
an initial vertical velocity of 40.0
m/s relative to the cart.
(a) How high does the rocket go?
(b) How far does the cart travel
while the rocket is in air?
(c) Where does the rocket land
relative to the cart?
Chapter 3, Problem 53
Your spaceship lands on an unknown planet. To
determine the local value of g, you ask a steel-toed
crew member to kick a stone, and you find that if
she kicks it at 17.6 m/s at various angles, the
maximum range she can achieve is 33.8 m.
(a) What is g on this planet?
(b) How long was the stone in the air?
(c) How high above the ground did the stone go?
Chapter 3 Problem 55
A baseball thrown at an angle of 60.0° above the
horizontal strikes a building 18.0 m away at a point
8.00 m above the point from which it was thrown.
Ignore air resistance.
(a) Find the magnitude of the initial velocity of the
baseball (the velocity with which the baseball is
thrown).
(b) Find the magnitude and direction of the velocity
of the baseball just before it strikes the building.
Chapter 11, Problem 30
A 2.00 kg frictionless block is attached to an ideal
spring with force constant 315 N/m. Initially the
spring is neither stretched nor compressed, but the
block is moving in the negative direction at 12.0 m/s.
Find
(a)the amplitude of the motion,
(b) the maximum acceleration of the block, and
(c) the maximum force the spring exerts on the block.
Chapter 11, Problem 31
Repeat the previous problem, but assume that
initially the block has velocity -4.00 m/s and
displacement +0.200 m.
Chapter 11, Problem 32
A tuning fork has the tip of each of its two
prongs vibrating in simple harmonic motion with a
frequency of 392 Hz and an amplitude of 0.600
mm, while a housefly with mass 0.0270 g is holding
onto the tip of one of the prongs. What are
(a) the maximum speed of the tip of a prong and
(b) the fly’s maximum kinetic energy?