4.166. A string 120 cm in length sustains a standing wave, with
the points of the string at which the displacement amplitude is equal
to 3.5 mm being separated by 15.0 cm. Find the maximum displacement amplitude. To which overtone do these oscillations correspond?
4.167. Find the ratio of the fundamental tone frequencies of two
identical strings after one of them was stretched by i = 2.0% and
the other, by 12 = 4.0%. The tension is assumed to 'be proportional
to the elongation.
4.168. Determine in what way and how many times will the fundamental tone frequency of a stretched wire change if its length is
shortened by 35% and the tension increased by 70%.
4.169. To determine the sound propagation velocity in air by
acoustic resonance technique one can use a pipe with a piston and
a sonic membrane closing one of its ends. Find the velocity of sound
if the distance between the adjacent positions of the piston at which
resonance is observed at a frequency v = 2000 Hz is equal to 1 =
= 8.5 cm.
4.170. Find the number of possible natural oscillations of air column in a pipe whose frequencies lie below v, = 1250 Hz. The length
of the pipe is 1 = 85 cm. The velocity of sound is v = 340 m/s.
Consider the two cases:
(a) the pipe is closed from one end;
(b) the pipe is opened from both ends.
The open ends of the pipe are assumed to be the antinodes of displacement.
4.171. A copper rod of length 1 = 50 cm is clamped at its midpoint.
Find the number of natural longitudinal oscillations of the rod in
the frequency range from 20 to 50 kHz. What are those frequencies
equal to?
4.172. A string of mass rn, is fixed at both ends. The fundamental
tone oscillations are excited with circular frequency co and maximum
displacement amplitude amax. Find:
(a) the maximum kinetic energy of the string;
(b) the mean kinetic energy of the string averaged over one oscillation period.
4.173. A standing wave = a sin kx•cos cot is maintained in a
homogeneous rod with cross-sectional area S and density p. Find the
total mechanical energy confined between the sections corresponding
to the adjacent displacement nodes.
4.174. A source of sonic oscillations with frequency vo = 1000 Hz
moves at right angles to the wall with a velocity u = 0.17 m/s.
Two stationary receivers R1 and R 2 are located on a straight line,
coinciding with the trajectory of the source, in the following succession: R1-source-R 2-wall. Which receiver registers the beatings and
what is the beat frequency? The velocity of sound is equal to v
= 340 m/s.
4.175. A stationary observer receives sonic oscillations from two
tuning forks one of which approaches, and the other recedes with
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the points of the string at which the displacement amplitude is equal
to 3.5 mm being separated by 15.0 cm. Find the maximum displacement amplitude. To which overtone do these oscillations correspond?
4.167. Find the ratio of the fundamental tone frequencies of two
identical strings after one of them was stretched by i = 2.0% and
the other, by 12 = 4.0%. The tension is assumed to 'be proportional
to the elongation.
4.168. Determine in what way and how many times will the fundamental tone frequency of a stretched wire change if its length is
shortened by 35% and the tension increased by 70%.
4.169. To determine the sound propagation velocity in air by
acoustic resonance technique one can use a pipe with a piston and
a sonic membrane closing one of its ends. Find the velocity of sound
if the distance between the adjacent positions of the piston at which
resonance is observed at a frequency v = 2000 Hz is equal to 1 =
= 8.5 cm.
4.170. Find the number of possible natural oscillations of air column in a pipe whose frequencies lie below v, = 1250 Hz. The length
of the pipe is 1 = 85 cm. The velocity of sound is v = 340 m/s.
Consider the two cases:
(a) the pipe is closed from one end;
(b) the pipe is opened from both ends.
The open ends of the pipe are assumed to be the antinodes of displacement.
4.171. A copper rod of length 1 = 50 cm is clamped at its midpoint.
Find the number of natural longitudinal oscillations of the rod in
the frequency range from 20 to 50 kHz. What are those frequencies
equal to?
4.172. A string of mass rn, is fixed at both ends. The fundamental
tone oscillations are excited with circular frequency co and maximum
displacement amplitude amax. Find:
(a) the maximum kinetic energy of the string;
(b) the mean kinetic energy of the string averaged over one oscillation period.
4.173. A standing wave = a sin kx•cos cot is maintained in a
homogeneous rod with cross-sectional area S and density p. Find the
total mechanical energy confined between the sections corresponding
to the adjacent displacement nodes.
4.174. A source of sonic oscillations with frequency vo = 1000 Hz
moves at right angles to the wall with a velocity u = 0.17 m/s.
Two stationary receivers R1 and R 2 are located on a straight line,
coinciding with the trajectory of the source, in the following succession: R1-source-R 2-wall. Which receiver registers the beatings and
what is the beat frequency? The velocity of sound is equal to v
= 340 m/s.
4.175. A stationary observer receives sonic oscillations from two
tuning forks one of which approaches, and the other recedes with
191