vessel in form of long cylinder was being filled from tap water at uniform rate. a man observe that there were resonance at intervals of 100s with a tuning fork of frequency 300 hz. calculate the rate of supply of water in cm^3 per second.given radius=.10m and velocity of sound=330

To solve the problem of calculating the rate of supply of water into a cylindrical vessel that resonates with a tuning fork, we need to understand the relationship between the frequency of the sound, the speed of sound in water, and the dimensions of the cylinder. Let's break this down step by step.

Understanding Resonance in the Cylinder

When the tuning fork produces a sound at a frequency of 300 Hz, it creates standing waves in the cylindrical vessel. The resonance occurs at specific intervals, which in this case is every 100 seconds. This means that the length of the water column in the cylinder changes in such a way that it supports these standing waves.

Calculating the Wavelength

The speed of sound in water is given as 330 m/s. The relationship between speed (v), frequency (f), and wavelength (λ) is given by the formula:

v = f × λ

Rearranging this gives us:

λ = v / f

Substituting the values:

• v = 330 m/s
• f = 300 Hz

Now, we can calculate the wavelength:

λ = 330 m/s / 300 Hz = 1.1 m

Determining the Length of the Water Column

In a cylindrical vessel, resonance occurs at specific lengths of the water column. The fundamental frequency (first harmonic) corresponds to a length equal to half the wavelength:

L = λ / 2 = 1.1 m / 2 = 0.55 m

Since the resonance occurs every 100 seconds, we can infer that the water level rises to this length in that time frame.

Calculating the Volume of Water Supplied

The volume of water in the cylinder can be calculated using the formula for the volume of a cylinder:

V = πr²h

Where:

• r = radius of the cylinder = 0.10 m
• h = height of the water column = 0.55 m

Now, substituting the values into the volume formula:

V = π × (0.10 m)² × 0.55 m

V = π × 0.01 m² × 0.55 m ≈ 0.0173 m³

To convert this volume into cubic centimeters (cm³), we use the conversion factor (1 m³ = 1,000,000 cm³):

V ≈ 0.0173 m³ × 1,000,000 cm³/m³ ≈ 17300 cm³

Finding the Rate of Supply of Water

Since this volume is supplied in 100 seconds, we can now calculate the rate of supply:

Rate = Volume / Time = 17300 cm³ / 100 s = 173 cm³/s

Final Result

The rate of supply of water into the cylindrical vessel is approximately 173 cm³ per second.