Advertisements
Advertisements
Question
A 660 Hz tuning fork sets up vibration in a string clamped at both ends. The wave speed for a transverse wave on this string is 220 m s−1 and the string vibrates in three loops. (a) Find the length of the string. (b) If the maximum amplitude of a particle is 0⋅5 cm, write a suitable equation describing the motion.
Advertisements
Solution
Given:
Frequency (f) = 660 Hz
Wave speed (v) = 220 m/s
\[\text{ Wave length,} \lambda = \frac{v}{f} = \frac{220}{660} = \frac{1}{3} m\]
(a) No. of loops, n = 3
∴ \[L = \frac{n}{2}\lambda\]
\[\Rightarrow L = \frac{3}{2} \times \frac{1}{3}\]
\[ \Rightarrow L = \frac{1}{2} m = 50 \text{ cm }\]
(b) Equation of resultant stationary wave can be given by:
\[y = 2A\cos\left( \frac{2\pi x}{\lambda} \right)\sin\left( \frac{2\pi vL}{\lambda} \right)\]
\[ \Rightarrow y = 0 . 5 \cos\left( \frac{2\pi x}{\frac{1}{3}} \right)\sin\left( \frac{2\pi \times 220 \times t}{\frac{1}{3}} \right)\]
\[ \Rightarrow y = 0 . 5 cm \cos\left( 6\pi x m^{- 1} \right) \sin\left( 1320\pi t s^{- 1} \right)\]
APPEARS IN
RELATED QUESTIONS
When longitudinal wave is incident at the boundary of denser medium, then............................
- compression reflects as a compression.
- compression reflects as a rarefaction.
- rarefaction reflects as a compression.
- longitudinal wave reflects as transverse wave.
A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + π/4)
Where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave?
If it is travelling, what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c) What is the initial phase at the origin?
(d) What is the least distance between two successive crests in the wave?
Explain the reflection of transverse and longitudinal waves from a denser medium and a rared medium.
A wave going in a solid
(a) must be longitudinal
(b) may be longitudinal
(c) must be transverse
(d) may be transverse.
A wave moving in a gas
A steel wire of length 64 cm weighs 5 g. If it is stretched by a force of 8 N, what would be the speed of a transverse wave passing on it?
A vertical rod is hit at one end. What kind of wave propagates in the rod if (a) the hit is made vertically (b) the hit is made horizontally?
Two wires of different densities but same area of cross section are soldered together at one end and are stretched to a tension T. The velocity of a transverse wave in the first wire is double of that in the second wire. Find the ratio of the density of the first wire to that of the second wire.
A transverse wave described by \[y = \left( 0 \cdot 02 m \right) \sin \left( 1 \cdot 0 m^{- 1} \right) x + \left( 30 s^{- 1} \right)t\] propagates on a stretched string having a linear mass density of \[1 \cdot 2 \times {10}^{- 4} kg m^{- 1}\] the tension in the string.
A circular loop of string rotates about its axis on a frictionless horizontal place at a uniform rate so that the tangential speed of any particle of the string is ν. If a small transverse disturbance is produced at a point of the loop, with what speed (relative to the string) will this disturbance travel on the string?
A heavy but uniform rope of length L is suspended from a ceiling. (a) Write the velocity of a transverse wave travelling on the string as a function of the distance from the lower end. (b) If the rope is given a sudden sideways jerk at the bottom, how long will it take for the pulse to reach the ceiling? (c) A particle is dropped from the ceiling at the instant the bottom end is given the jerk. Where will the particle meet the pulse?
A transverse wave of amplitude 0⋅50 mm and frequency 100 Hz is produced on a wire stretched to a tension of 100 N. If the wave speed is 100 m s−1, what average power is the source transmitting to the wire?
A tuning fork of frequency 440 Hz is attached to a long string of linear mass density 0⋅01 kg m−1 kept under a tension of 49 N. The fork produces transverse waves of amplitude 0⋅50 mm on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string?
If the speed of a transverse wave on a stretched string of length 1 m is 60 m−1, what is the fundamental frequency of vibration?
A wire, fixed at both ends is seen to vibrate at a resonant frequency of 240 Hz and also at 320 Hz. (a) What could be the maximum value of the fundamental frequency? (b) If transverse waves can travel on this string at a speed of 40 m s−1, what is its length?
The equation of a standing wave, produced on a string fixed at both ends, is
\[y = \left( 0 \cdot 4 cm \right) \sin \left[ \left( 0 \cdot 314 {cm}^{- 1} \right) x \right] \cos \left[ \left( 600\pi s^{- 1} \right) t \right]\]
What could be the smallest length of the string?
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a traveling wave, (ii) a stationary wave or (iii) none at all:
y = 2 cos (3x) sin (10t)
