1. Derive expressions for the kinetic and potential energies of a simple harmonic oscillator. Hence show that the total energy is conserved in S.H.M. in which positions of the oscillator, is the energy wholly kinetic or wholly potential?
2. Find the total energy of the particle executing S.H.M. and show graphically the variation of potential energy and kinetic energy with time in S.H.M. What is the frequency of these energies with respect to the frequency of the particle executing S.H.M.?
3. Discuss the Newton's formula for velocity of sound in air. What correction was applied to it by Laplace and why?
4. What are standing waves? Desire and expression for the standing waves. Also define the terms node and antinode and obtain their positions. 5. Discuss the formation of harmonics in a stretched string. Show that in
case of a stretched string the first four harmonics are in the ratio 1:2:3:4, 6. Give the differences between progressive and stationary waves.
7. Give a qualitative discussion of the modes of vibrations of a stretched string fixed at both the ends.
8. Give a qualitative discussion of the different modes of vibration of an open organ pipe.
9. Describe the various modes of vibrations of a closed organ pipe. 10. What are beats? How are they produced? Briefly discuss one application
for this phenomenon.
11. State Doppler's effect in sound obtain an expression for apparent frequency when source and listener move away from each other
NUMERICALS
its displacement be 1
2of its amplitude..
2. A point describes SHM in a line 6 cm long. Its velocity, when passing through the centre of line is 18 cm s–1. Find the time period.
3. Find the period of vibrating particle (SHM), which has accelesation of 45 cm s–2, when displacement from mean position is 5 cm
4. A 40 gm mass produces on extension of 4 cm in a vertical spring. A mass of 200 gm is suspended at its bottom and left pulling down. Calculate the frequency of its vibration.
5. The acceleration due to gravity on the surface of the moon is 1.7 ms–2.
What is the time period of a simple pendulum on the moon, if its time period on the earth is 3.5 s? [g = 9.8 ms–2]
6. Calculate the energy possessed by stone of mass 200 g executing S.H.M of amplitude 1 cm and time period 4s.
7. A particle executes S.H.M of amplitude 25 cm and time period 3s. What is the minimum time required for the particle to move between two points 12.5 cm on eitherside of the mean position?
8. The vertical motion of a huge piston in a machine is approximately S.H.M with a frequency of 0.5 s–1. A block of 10kg is placed on the piston. What
is the maximum amplitude of the piston's S.H.M. for the block and piston to remain together?
9. At what temperature will the speed of sound be double its value at 273°K? 10. A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this spring, when displaced and released, oscillates with a period of 0.60 s. What is the weight of the body? 11. A steel wire 80 cm long has a mess 8 mg. If the wire is under tension of
400 N, what is the speed of transverse waves in the wire?
12. You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation, Estimate the values of (a) the spring constant and (b) the damping constant ‘b’ for the spring and shock absorber system of one wheel assuming that each wheel supports 750 kg.
13. A string of mass 2.5 kg is under a tension of 200N. The length of the stretched string is 20m. If a transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end? 14. A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should
be the tension in the wire so that the speed of a transverse wave on the wire equals the speed of sound in day air at 20°C (is equal to 343 ms–1)?
15. The equation of a plane progressive wave is given by the equation y = 10 sin 2 (t – 0.005x) where y and x are in cm and t in seconds. Calculate the amplitude, frequency, wave length and velocity of the wave. 16. Find the frequency of note emitted (fundamental note) by a string 1m long and stretched by a load of 20 kg, if this string weighs 4.9 g. Given, g = 980 cms–2
17. A pipe 20 cm long is closed at one end, which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will this same source can be in resonance with the pipe, if both ends are open? Speed of sound = 340 ms–1
18. One end of a long string of linear mass density 8.0 × 10–3 kg m–1 is
connected to an electrically driven tunning folk of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At t = 0, the left end of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive x direction. The amplitude of wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describes the wave on the string.
19. The transverse displacement of a string (clamped at its two ends) is given by
y(x,t) = 0.06 sin2 3
x cos(120t)
where x,y are in m and t is in s. The length of the string is 1.5 m and its mass is 3.0 × 10–2kg. Answer the following.
(a) Does the function represent a travelling or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in
opposite directions. What are the wavelength frequency and speed of propagation of each wave?
(c) Determine the tension in the string.
20. A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency 45 Hz. The mass of the wire is 3.5 × 10–2 kg and
its linear density is 4.0 × 10–2 kg m–. What is (a) the speed of transverse
wave on the string and (b) the tension in the string?
21. A steel sod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod as given to be 2.53 kHz. What is the speed of sound in steel?