Advertisements
Advertisements
प्रश्न
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?
उत्तर
(a) Let m be the mass per unit length of the string.
Consider an element at a distance x from the lower end.
Here,
Weight acting downwards = (mx)g
∴ Tension in the string at the upper part = mgx
The velocity of transverse vibration is given as
\[v = \sqrt{\left( \frac{T}{m} \right)} = \sqrt{\left( \frac{mgx}{m} \right)}\]
\[ \Rightarrow v = \sqrt{\left( gx \right)}\]
(b) Let the time taken be dt for the small displacement dx.
Thus, we have:
\[dt = \frac{dx}{v} = \frac{dx}{\sqrt{\left( gx \right)}}\]
\[\therefore \text{ Total time, } T = \int\limits_0^L \frac{dx}{\sqrt{\left( gx \right)}} = \sqrt{\left( \frac{4L}{g} \right)}\]
(c) Suppose after time t, the pulse meets the particle at a distance y from the lower end of the rope.
Now,
\[t = \int\limits_0^y \frac{dx}{\sqrt{\left( gx \right)}}\]
\[ = \sqrt{\left( \frac{4y}{g} \right)}\]
∴ Distance travelled by the particle in this time, S = \[L - y\]
Using the equation of motion, we get:
\[S = ut + \frac{1}{2} g t^2 \]
\[ \Rightarrow L - y = \left( \frac{1}{2} \right) g \times \left\{ \left( \sqrt{\frac{4y}{g}} \right)^2 \right\}\]
\[ \Rightarrow L - y = 2y\]
\[ \Rightarrow 3y = L\]
\[ \Rightarrow y = \frac{L}{3}\]
Thus, the particle will meet the pulse at a distance
\[\frac{L}{3}\] from the lower end.
APPEARS IN
संबंधित प्रश्न
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?
Explain why (or how): Bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
Explain why (or how) Solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases
Explain the reflection of transverse and longitudinal waves from a denser medium and a rared medium.
You are walking along a seashore and a mild wind is blowing. Is the motion of air a wave motion?
A mechanical wave propagates in a medium along the X-axis. The particles of the medium
(a) must move on the X-axis
(b) must move on the Y-axis
(c) may move on the X-axis
(d) may move on the Y-axis.
A wave moving in a gas
A particle on a stretched string supporting a travelling wave, takes 5⋅0 ms to move from its mean position to the extreme position. The distance between two consecutive particles, which are at their mean positions, is 2⋅0 cm. Find the frequency, the wavelength and the wave speed.
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?
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.
Consider the following statements about sound passing through a gas.
(A) The pressure of the gas at a point oscillates in time.
(B) The position of a small layer of the gas oscillates in time.
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.
In the arrangement shown in figure , the string has a mass of 4⋅5 g. How much time will it take for a transverse disturbance produced at the floor to reach the pulley? Take g = 10 m s−2.
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 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.
