an related ambiguity, because tan() goes through two complete cycles
in the range (, ]. This means that the four permutations of the
signs: x0, v0 produce the same angle twice when evaluating as an
inverse tangent, and all four of them produce an amplitud
British to be mad. Roman soldiers also discovered another important
aspect of resonance it can destroy humanengineered structures! The
standard marching pace of the Roman soldiers was 4000 paces per
hour, just over two steps per second. This pace could ea
this chapter after the real solultion is obtained). There is a price you will
pay if you do. You will never understand where the solution comes from
or how to solve the slightly more difficult damped SHO problem, and
will therefore have to memorize the so
of pressure controls in its neck to reduce this pressure so that it doesnt
have a brain aneurism every time it gets thirsty! Giraffes, like humans
and most other large animals, have a second problem. The heart
doesnt maintain a steady pressure differentia
energy of an oscillator is found by integratng: U(x) = Z x 0 kx dx = 1
2 kx2 = 1 2 kA2 cos2 (t + ) (789) if we use the (usual but not
necessary) conventon that U(0) = 0 when the mass is at the equilibrium
displacement, x = 0. The total mechanical energy i
the equilibrium bond length), and r is along the bond axis. This function
is portrayed as the solid line in figure 130. An alternative is the Morse
potential 198 (energy): UM(r) = Umin 1 e a(rrb) 2 (917) In this
expression Umin and rb have the same meanin
with Lennard-Jones or Morse potentials (or any of a number of other
related forms with more or less virtue for any particular problem) and
sacrifice precision in the result for computability. In our case, even these
relatively simple effective potentials
separation of the atoms as a. or structured solids. This is illustrated in
figure 131, which is basically a mental cartoon model for a generic solid
lots of atoms in a regular cubic lattice with a cube side a, where the
interactomic forces that hold each
you will just get the first solution, which is wrong! You have to choose
between: x(t) = 0.1 2 cos(10t + 4 ) (844) and x(t) = 0.1 2 cos(10t
3 4 ) (845) (and the corresponding v(t)s). Either one is a valid solution.
Again, there is no particularly compell
trick is to use the Taylor series for the cosine function: cos() = 1 2 2!
+ 4 4! + . (879) and keep only the first term: 1 cos() = 2 2! + 4
4! + . 2 2 = 1 2 2 cos2 (t + ) (880) You should now be able to
see that in fact, the total energy of the oscillator
The amplitude damps exponentially as time advances. After a certain
amount of time, the amplitude is halved. After the same amount of
time, it is halved again. 416 Week 9: Oscillations The frequency is
shifted so that it is smaller than 0, the frequency o
earth has a certain period T. A physicist on the moon, where the
acceleration near the surface is around g/6, wants to make a pendulum
with the same period. What mass mm and length Lm of string could be
used to accomplish this? Week 9: Oscillations 439 Pr
multiply this by one in the form Nx Nx : F a2 A = keff Nx Nx x (926) We
multiply this by one in the form L=aNx L=aNx : F a2 A = keffa Nxx Nxa
= keffa L L (927) Week 9: Oscillations 431 x a L L F F A Figure 132:
The same lattice stretched by an amount L as
concludes our general discussion of simple harmonic oscillators in the
specific context of a mass on a spring. However, there are many more
systems that oscillate harmonically, or nearly harmonically. Lets study
another very important one next. 9.2: The P
potential UL(r) and the dashed line is the Morse potential UM(r). Note
that the two are very closely matched for short range repulsion, but the
Morse potential dies off faster than the 1/r6 London form expected at
longer range. force is the negative slope
function y(x, t) = f(x vt) (961) satisfies the wave equation. Any shape
of wave created on the string and propagating to the right or left is a
solution to the wave equation, although not all of these waves will
vanish at the ends of a string. 452 Week 10
undergrads usually have learned by the time they take this course.
9.4.2: Solution to Damped, Driven, Simple Harmonic Oscillator We will
not, actually, fully develop this solution this semester. It is a bit easier to
do so in the context of LRC circuits n
of the string dE/dx in this formulation: dE dx = 1 2 2A 2 cos2 (kx t)
+ 1 2 T k2A 2 cos2 (kx t) = 1 2 2A 2 cos2 (kx t) + cos2 (kx t)
= 2A 2 cos2 (kx t) (1004) because 2 = T k2 from T = 2 k2 = v
2 . There is apparently exactly as much kinetic energy as the
tension typically other parts of the skeletal structure tendons or
cartilage in the joints fail before the actual bone does in these
situations. Bone is basically brittle and easy to chip, but does have a
significant degree of compressive, tensile, or she
values for a fishing bobber). 442 Week 9: Oscillations Problem 7. L M
max This problem will help you learn required concepts such as:
Simple Harmonic Oscillation Torque Newtons Second Law for
Rotation Moments of Inertia Small Angle Approximation so pleas
x=x0 x 2 + . = U0 + 1 2 keffx 2 (922) where keff = d 2U dx2 x=x0
(923) What this means is that any mass at a stable equilibrium point
will usually behave like a mass on a spring for motion close to the
equilibrium point! If pulled a short distance away an
medium with damping coefficient b. (Gravity, if present at all, is
irrelevant as shown in class). The net force on the mass when displaced
by x from equilibrium and moving with velocity vx is thus: Fx = max =
kx bvx (in one dimension). a) Convert this equ
. (7.7) Where, R = resistance of the
resistor = resistivity of the resistor L = the length of the
resistor A = average cross-sectional area Combining Eq. (7.6)
and Eq. (7.7), we get I = R V x R I = 1 x L A x V I = 1 x A x L
V2 - V1 . (7.8) Where, 1 = elec
+ rt) . (5.19) (ii) At constant volume Vo, the pressure
becomes pt = Po(1+ t) The product PV then becomes PoVo(1 +
t) (5.20) If Boyles law is obeyed, all values of the
product PV at the same temperature to C must be the same.
PoVo(1 + rt) = PoVo(1 + t) .
Pressure versus Temperature PHY 113 c Our conclusion from
this graph is that the pressure on the gas varies linearly with the
temperature. When the graph is extrapolated, the pressure Po
at 0o C can be read from the graph. Further extrapolation
produces t
gas laws explain the gas laws through the use of graphs
distinguish between a real gas and an ideal gas express the
equation of state of an ideal gas solve problems on these gas
laws. 3.0 MAIN CONTENT 3.1 GAS LAWS You will recall that four
properties are
. (8.2) k is the constant of
proportionality whose value depends on two factors (the
negative sign shows that heat is lost to the surrounding): The
nature of the exposed surface of the material surface
emmisivitiy Its surface area A - t Q = A ( - o)
. (8.
process is described as an isothermal expansion; When a gas
is allowed to expand without heat entering or leaving the gas,
the gas is said to undergo an adiabatic expansion; Work is
done with rise in internal energy. ANSWER TO SELF
ASSESSMENT EXERCISE Fro
and Archimedes Principle to explain convection in fluids.
Newtons law of cooling was used to quantify the rate of loss of
heat under natural or forced condition, which is proportional to
the excess temperature of its temperature and its surrounding
temper