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 the same source be in resonance with the pipe if both ends are open? (Speed of sound in air is 340 m s-1).
First (Fundamental); No
Length of the pipe, l = 20 cm = 0.2 m
Source frequency = nth normal mode of frequency, Vn = 430 Hz
Speed of sound, v = 340 m/s
In a closed pipe, the nth normal mode of frequency is given by the relation:
Vn = (2n -1) v / 4l ; n is an integer = 0, 1, 2, 3...
430 = (2n-1) 340 / (4x0.2)
2n -1 = (430 x 4 x 0.2) / 340 = 1.01
2n = 2.01
n ≈ 1
Hence, the first mode of vibration frequency is resonantly excited by the given source.
In a pipe open at both ends, the nth mode of vibration frequency is given by the relation:
Vn = nv / 2l
n = 2lVn / v
= 2 x 0.2 x 430 / 340 = 0.5
Since the number of the mode of vibration (n) has to be an integer, the given source does not produce a resonant vibration in an open pipe.
State the number of significant figures in the following:
(a) 0.007 m2
(b) 2.64 x 1024 kg
(c) 0.2370 g cm-3
(d) 6.320 J
(e) 6.032 N m-2
(f) 0.0006032 m2
Fill in the blanks by suitable conversion of units:
(a) 1 kg m2s–2= ....g cm2 s–2
(b) 1 m =..... ly
(c) 3.0 m s–2=.... km h–2
(d) G = 6.67 × 10–11 N m2 (kg)–2=.... (cm)3s–2 g–1.
A physical quantity P is related to four observables a, b, c and d as follows :
The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ?
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The mass of a box measured by a grocer's balance is 2.300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is
(a) the total mass of the box,
(b) the difference in the masses of the pieces to correct significant figures?
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y(x,t) = 3.0 sin [36t + 0.018x + 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?
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(a) climbs up with an acceleration of 6 m s-2
(b) climbs down with an acceleration of 4 m s-2
(c) climbs up with a uniform speed of 5 m s-1
(d) falls down the rope nearly freely under gravity?
(Ignore the mass of the rope).
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s–1 can go without hitting the ceiling of the hall?
A physical quantity P is related to four observables a, b, c and d as follows :
The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ?
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