Derive the approximate form of Heisenberg's uncertainty principle for energy and time, Δ E Δ t ≈ h , using the following arguments: Since the position of a particle is uncertain by Δ x ≈ λ , where λ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse Δ x . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to λ . Find Δ t and Δ E ; then multiply them to give the approximate uncertainty principle.
Derive the approximate form of Heisenberg's uncertainty principle for energy and time, Δ E Δ t ≈ h , using the following arguments: Since the position of a particle is uncertain by Δ x ≈ λ , where λ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse Δ x . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to λ . Find Δ t and Δ E ; then multiply them to give the approximate uncertainty principle.
Derive the approximate form of Heisenberg's uncertainty principle for energy and time,
Δ
E
Δ
t
≈
h
, using the following arguments: Since the position of a particle is uncertain by
Δ
x
≈
λ
, where
λ
is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse
Δ
x
. Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to
λ
. Find
Δ
t
and
Δ
E
; then multiply them to give the approximate uncertainty principle.
Portfolio Problem 3. A ball is thrown vertically upwards with a speed vo
from the floor of a room of height h. It hits the ceiling and then returns to the
floor, from which it rebounds, managing just to hit the ceiling a second time.
Assume that the coefficient of restitution between the ball and the floor, e, is
equal to that between the ball and the ceiling. Compute e.
Portfolio Problem 4. Consider two identical springs, each with natural length
and spring constant k, attached to a horizontal frame at distance 2l apart. Their
free ends are attached to the same particle of mass m, which is hanging under
gravity. Let z denote the vertical displacement of the particle from the hori-
zontal frame, so that z < 0 when the particle is below the frame, as shown in
the figure. The particle has zero horizontal velocity, so that the motion is one
dimensional along z.
000000
0
eeeeee
(a) Show that the total force acting on the particle is
X
F-mg k-2kz 1
(1.
l
k.
(b) Find the potential energy U(x, y, z) of the system such that U
x = : 0.
= O when
(c) The particle is pulled down until the springs are each of length 3l, and then
released. Find the velocity of the particle when it crosses z = 0.
In the figure below, a semicircular conductor of radius R = 0.260 m is rotated about the axis AC at a constant rate of 130 rev/min. A uniform magnetic field of magnitude 1.22 T fills the entire region below the axis and is directed out of the page.
R
Pout
(a) Calculate the maximum value of the emf induced between the ends of the conductor.
1.77
v
(b) What is the value of the average induced emf for each complete rotation?
0
v
(c) How would your answers to parts (a) and (b) change if the magnetic field were allowed to extend a distance R above the axis of rotation? (Select all that apply.)
The value in part (a) would increase.
The value in part (a) would remain the same.
The value in part (a) would decrease.
The value in part (b) would increase.
The value in part (b) would remain the same.
The value in part (b) would decrease.
×
(d) Sketch the emf versus time when the field is as drawn in the figure. Choose File No file chosen
This answer has not been graded yet.
(e) Sketch the emf…
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