A uniform rope of length L and mass m₁ hangs vertically from a rigid support. A block of mass m₂ is attached to the free end of the rope. A transverse pulse of wavelength $$\lambda $$₁ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is $$\lambda $$₂. The ratio $$\lambda $$₂/$$\lambda $$₁ is

A string is stretched between fixed points separated by 75.0 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz. There are no other resonant frequencies between these two. The lowest resonant frequency for this string is

4.0 g of a gas occupies 22.4 litres at NTP. The specific heat capacity of the gas at constant volume is 5.0 J K^$$-$$1 mol^$$-$$1. If the speed of sound in this gas at NTP is 952 m s^$$-$$1, then the heat capacity at constant pressure is
(Take gas constant R $$=$$ 8.3 J K^$$-$$1 mol^$$-$$1)

A source of sound S emitting waves of frequency 100 Hz and an observer O are located at some distance from each other. The source is moving with a speed of 19.4 m s^$$-$$1 at an angle of 60^o with the source observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer (velocity of sound in air 330 m s^$$-$$1), is