Sound, Doppler Effect & Fluid Mechanics

Density measures how much mass is contained in a given volume of a substance. It is calculated by dividing the mass of the material by its volume, providing insight into how compact or spread out the matter is within a specific space. This relationship helps in characterizing materials, predicting buoyancy, and understanding various physical properties in science and engineering.

Bernoulli's principle states that in a fluid flow, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This relationship is due to the conservation of energy in a flowing fluid. When the fluid moves faster, it has less time to exert pressure on its surroundings, resulting in lower static pressure at points of higher velocity. Thus, at the same height, higher fluid speed indeed correlates with lower static pressure, confirming the statement as true.

According to the principle of conservation of mass, when a fluid flows through a pipe that narrows, its speed increases while the cross-sectional area decreases. Using the equation of continuity, A1V1 = A2V2, where A is the area and V is the velocity, we can solve for the new speed. Given the initial area (6.0 cm²) and speed (2.0 m/s), and the narrowed area (2.0 cm²), we find that the new speed (V2) must be 6.0 m/s to maintain the flow rate.

According to the continuity equation, the product of the cross-sectional area of a pipe and the fluid velocity must remain constant for incompressible flow. When a pipe narrows, the cross-sectional area decreases, which means that to maintain the same flow rate, the fluid speed must increase. This relationship illustrates the principle of conservation of mass in fluid dynamics, where a reduction in area necessitates an increase in velocity to ensure that the same volume of fluid passes through the pipe over time.

An object that is 75% submerged in water displaces a volume of water equal to 75% of its own volume. According to Archimedes' principle, the weight of the displaced water equals the weight of the object. Since the density of water is 1000 kg/m³, the weight of the displaced water can be calculated. If 75% of the object's volume displaces water weighing 1000 kg/m³, the density of the object must be 750 kg/m³ to balance the forces, as it is less dense than water but still significantly heavy enough to be submerged.

Archimedes' principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This means that when an object is placed in a fluid, it pushes aside a volume of fluid equal to its own volume, and the weight of this displaced fluid is what generates the buoyant force. Therefore, the buoyant force acting on the object is directly related to the weight of the fluid that has been displaced, not the weight of the object itself.

Pascal's principle asserts that when pressure is applied to a confined fluid, the change in pressure is distributed uniformly throughout the fluid. This means that any increase in pressure at one point in the fluid is felt equally at all other points, regardless of the shape or size of the container. This principle is fundamental in understanding how hydraulic systems operate, where a small force applied in one area can create a larger force in another area due to this equal transmission of pressure.

Gauge pressure can be calculated using the formula \( P = \rho g h \), where \( P \) is the gauge pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity, and \( h \) is the depth. For fresh water at a depth of 5.0 m, substituting the values gives \( P = 1000 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \times 5.0 \, \text{m} = 49,000 \, \text{Pa} \) or 49 kPa. Thus, the gauge pressure at this depth is 49 kPa.

Gauge pressure measures the pressure of a fluid relative to the ambient atmospheric pressure. It indicates how much greater the pressure is compared to atmospheric pressure, meaning it does not include the atmospheric pressure itself in its measurement. Therefore, when gauge pressure is reported, it reflects only the excess pressure above the surrounding atmospheric level, confirming that the statement is true.

Specific gravity is the ratio of the density of a substance to the density of water. Given a specific gravity of 0.80, we can calculate the density of the substance using the formula: density of substance = specific gravity × density of water. Substituting the values, we get: density = 0.80 × 1000 kg/m³ = 800 kg/m³. Thus, the substance's density is 800 kg/m³.

The Doppler effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. When the source approaches the observer, the waves compress, resulting in a higher frequency (or pitch). Conversely, as the source moves away, the waves stretch, leading to a lower frequency. This phenomenon is commonly experienced with sound waves, such as the changing pitch of a passing siren. It is crucial in various fields, including astronomy, where it helps determine the movement of stars and galaxies.

Intensity of sound or light from a point source decreases with the square of the distance from the source, following the inverse square law. The formula for intensity at a distance \( r \) is \( I = \frac{P}{4\pi r^2} \), where \( P \) is the power. Given the intensity at 4 m is 3.0 W/m², we can calculate the intensity at 12 m. The ratio of the distances squared (from 4 m to 12 m) is \( \frac{4^2}{12^2} = \frac{16}{144} = \frac{1}{9} \). Thus, the new intensity is \( 3.0 \times \frac{1}{9} = 0.33 \) W/m².

When the distance from a point sound source is tripled, the intensity of the sound decreases with the square of the distance. This is described by the inverse square law, which states that intensity is inversely proportional to the square of the distance from the source. Therefore, if the distance is increased by a factor of three, the intensity becomes 1/(3^2) = 1/9 of the original intensity.

The intensity of sound from a point source decreases with the square of the distance from the source. This relationship is described by the formula I = P/(4πr²), where I represents intensity, P is the power of the source, and r is the distance from the source. As the distance increases, the sound energy spreads over a larger area, resulting in lower intensity. The 4π factor accounts for the spherical shape of the wavefronts emanating from the point source, illustrating how sound intensity diminishes with increased distance.

When a sound source moves away from an observer, the sound waves are stretched, leading to a phenomenon known as the Doppler effect. This stretching causes the observed frequency of the sound to decrease, making it lower than the actual frequency emitted by the source. As a result, the observer perceives a lower pitch as the source recedes, confirming that the statement is true.

When a listener approaches a sound source, the observed frequency increases due to the Doppler effect. The formula for calculating the observed frequency (f') when the source is stationary and the listener is moving towards it is given by f' = f * (v + v₀) / v, where f is the source frequency, v is the speed of sound, and v₀ is the speed of the listener. Substituting the values (f = 680 Hz, v = 340 m/s, v₀ = 15 m/s) into the formula results in an observed frequency of approximately 710 Hz.

When a sound source approaches an observer, the observed frequency increases due to the Doppler effect. The formula for the observed frequency \( f' \) is given by \( f' = f \frac{v + v_0}{v - v_s} \), where \( f \) is the source frequency, \( v \) is the speed of sound, \( v_0 \) is the speed of the observer (0 if stationary), and \( v_s \) is the speed of the source. Plugging in the values (500 Hz, 340 m/s, and 20 m/s), we calculate the observed frequency to be 531 Hz.

When a sound source moves toward an observer, the observed frequency increases due to the Doppler effect. The formula f′ = f[v/(v − vₛ)] accounts for this by considering the speed of sound (v) and the speed of the source (vₛ) approaching the observer. As the source moves closer, the sound waves are compressed, resulting in a higher frequency perceived by the observer. This formula effectively captures the relationship between the source's motion and the change in frequency.

When a sound source approaches an observer, the sound waves are compressed, leading to a decrease in wavelength and an increase in frequency. This phenomenon, known as the Doppler effect, causes the observer to perceive a higher pitch as the source gets closer. Consequently, the frequency of the sound observed is greater than the frequency emitted by the source, resulting in a higher perceived frequency.