A solid copper object hangs at the bottom of a

steel wire of negligible mass. The top end of the wire is
fixed. When the wire is struck, it emits sound with a fundamental
frequency of 300 Hz. The copper object is then submerged
in water so that half its volume is below the water
line. Determine the new fundamental frequency.
is this the question?

It is the question if you say it is.

Partially submerging the copper object will reduce the tension in the steel wire. There will be a buoyancy force equal to half the volume of the copper object multiplied by the density of water. Let V be the volume of the copper and g be the acceleration of gravity. The specific gravity of copper is 8.92, and that of water is 1.00.

Copper object weight = 8.92 V*g
Buoyancy force = 1.00*(V/2)*g = 0.50 V*g
Effective weight = 8.42 V*g
The tension T gets multiplied by a factor 8.42/8.92 = 0.9439.

The speed of waves in the wire AND the fundamental frequency get multiplied by the square root of this factor, or 0.9716.

The new fundamental frequency becomes
300*0.9716 = 291.5 Hz
That is about half of a half-tone on the musical scale.

Yes, that is the question. To determine the new fundamental frequency, we need to consider the change in the effective length and mass of the vibrating part of the wire.

When the copper object hangs at the bottom of the steel wire, the effective length of the vibrating part is the length of the wire. The fundamental frequency is given by the formula:

f1 = (1/2L1) * √(T/μ)

Where:
f1 is the fundamental frequency,
L1 is the effective length of the vibrating part of the wire,
T is the tension in the wire,
μ is the linear mass density of the wire.

Now, when the copper object is submerged in water, it displaces water equal to half of its volume. This means that the weight of the copper object is reduced by the weight of the water it displaces.

To find the new fundamental frequency, we need to calculate the new effective length and mass of the vibrating part of the wire:

1. Effective Length (L):
When the copper object is submerged, the effective length of the vibrating part is the length from the fixed end of the wire to the waterline.

2. Mass (m):
The linear mass density (μ) of the wire remains the same. However, the mass of the copper object hanging at the end of the wire increases due to the weight of the water it displaces.

After finding the new effective length and mass, we can use the same formula to calculate the new fundamental frequency:

f2 = (1/2L2) * √(T/μ)

Where:
f2 is the new fundamental frequency,
L2 is the new effective length of the vibrating part of the wire,
T is the tension in the wire,
μ is the linear mass density of the wire.

By plugging in the values for L2 and solving the equation, we can determine the new fundamental frequency.

Yes, that is the question. To determine the new fundamental frequency of the sound emitted when the copper object is submerged in water, we can use the principle of resonance.

The fundamental frequency of a vibrating object is determined by the length, material, and tension of the string or wire producing the sound. In this case, the length and tension of the wire are fixed, but the material changes when the copper object is submerged in water.

When the copper object is submerged in water, it experiences a buoyant force due to the displacement of water. This buoyant force reduces the effective tension in the wire, resulting in a change in the fundamental frequency of the sound emitted.

The change in the fundamental frequency is directly proportional to the square root of the ratio of the tension in air (T_air) to the tension in water (T_water) and the square root of the density of air (ρ_air) to the density of water (ρ_water).

Let's calculate the new fundamental frequency:

1. Start with the initial fundamental frequency (f_initial) of 300 Hz.

2. Determine the ratio of tensions in air and water:
T_ratio = √(T_air / T_water)

3. Calculate the ratio of densities of air and water:
ρ_ratio = √(ρ_air / ρ_water)

4. Calculate the new fundamental frequency (f_new) using the following formula:
f_new = f_initial * T_ratio * ρ_ratio

By substituting the respective values, we can calculate the new fundamental frequency.