Study Guide
Physics 16 Problem Set 12 Solutions
-
University:
Amherst College -
Course:
PHYS 116 | Introductory Physics I: Mechanics Academic year:
2021
-
Views:
292
Pages:
2
Author:
Zara Mcneil
Physics 16 Problem Set 12 Solutions
Y&F Problems
15.18.
IDENTIFY:
SET UP:
EXECUTE:
the same
Apply
to determine the tension at different points of the rope.
From Example 15.3,
,
.
and
(a) The tension at the bottom of the rope is due to the weight of the load, and the speed is
as found in Example 15.3.
(b) The tension at the middle of the rope is (21.0
and the wave speed is
(c) The tension at the top of the rope is
and the speed is
(See Challenge Problem (15.82) for the effects of varying tension on the time it takes to send signals.)
EVALUATE: The tension increases toward the top of the rope, so the wave speed increases from the
bottom of the rope to the top of the rope.
15.41.
IDENTIFY: Compare
given in the problem to Eq.(15.28). From the frequency and wavelength
for the third harmonic find these values for the eighth harmonic.
(a) SET UP: The third harmonic standing wave pattern is sketched in Figure 15.41.
Figure 15.41
EXECUTE:
(b) Eq. (15.28) gives the general equation for a standing wave on a string:
so
(c) The sketch in part (a) shows that
Comparison of
given in the problem to Eq. (15.28) gives
(d)
So,
from part (c)
so
period
(e)
(f )
so
is the fundamental
and
EVALUATE: The wavelength and frequency of the standing wave equals the wavelength and
frequency of the two traveling waves that combine to form the standing wave. In the 8th harmonic the
frequency and wave number are larger than in the 3rd harmonic.
15.54.
IDENTIFY:
The maximum vertical acceleration must be at least SET UP:
EXECUTE:
and thus
Using
and
, this becomes
.
EVALUATE: When the amplitude of the motion increases, the maximum acceleration of a point on the
rope increases.
15.68.
IDENTIFY: The time between positions 1 and 5 is equal to
string is given by Eq.(15.9).
SET UP:
. The velocity of points on the
Four flashes occur from position 1 to position 5, so the elapsed time is
.
The figure in the problem shows that
. At point P the amplitude of the standing wave
is 1.5 cm.
EXECUTE: (a)
and
.
.
.
(b) The fundamental standing wave has nodes at each end and no nodes in between. This standing wave
has one additional node. This is the 1st overtone and 2nd harmonic.
(c)
.
(d) In position 1, point P is at its maximum displacement and its speed is zero. In position 3, point P is
passing through its equilibrium position and its speed is
.
(e)
and
.
EVALUATE: The standing wave is produced by traveling waves moving in opposite directions. Each
point on the string moves in SHM, and the amplitude of this motion varies with position along the
string.
15.35.
IDENTIFY:
Evaluate
SET UP:
EXECUTE:
and
and see if Eq.(15.12) is satisfied for
.
.
.
(a)
solution of Eq.(15.12),
.
so for
to be a
and
(b) A standing wave is built up by the superposition of traveling waves, to which the relationship
applies.
EVALUATE:
solutions to the wave equation.
15.21.
IDENTIFY:
is a solution of the wave equation because it is a sum of
For a point source,
and
.
SET UP:
EXECUTE:
(b)
(a)
, with
and
.
.
(c)
EVALUATE: These are approximate calculations, that assume the sound is emitted uniformly in all
directions and that ignore the effects of reflection, for example reflections from the ground.
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Physics 16 Problem Set 12 Solutions
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