To determine the new fundamental frequency, we need to consider the change in effective length of the wire after the copper object is submerged in water.
1. The fundamental frequency of a vibrating string or wire is inversely proportional to its effective length. When the copper object is at the bottom, the effective length of the wire is the distance between the top end (fixed) and the bottom end (where the copper object hangs).
2. When half of the copper object's volume is submerged in water, it displaces an equal volume of water. According to Archimedes' principle, the volume of water displaced is equal to the volume of the object submerged.
3. Since density = mass/volume, the density of the copper object is greater than that of water. Therefore, when submerged, it will experience a buoyant force that partially counteracts its weight. This results in a reduction in the effective length of the wire.
To determine the new fundamental frequency, we can use the equation:
f = (1/2L) * sqrt(T/μ),
where:
- f is the fundamental frequency,
- L is the effective length of the wire,
- T is the tension in the wire, and
- μ is the linear mass density of the wire.
Since the tension and linear mass density of the wire remain constant, we only need to calculate the change in effective length (L).
As the copper object is submerged by half its volume, we can assume it displaces an equal volume of water. The volume of water displaced is equal to the volume of the object submerged.
Now, let's assume that the initial length of the copper object hanging below the water line is L1. After half its volume is submerged, the length of the submerged part of the copper object is L1/2.
Therefore, the change in effective length is equal to the length of the submerged part of the copper object, i.e., L1/2.
The new effective length of the wire is the original length minus the change in effective length.
Hence, the new effective length is L - L1/2.
Finally, we can substitute the new effective length into the fundamental frequency formula to calculate the new fundamental frequency (f'):
f' = (1/2(L - L1/2)) * sqrt(T/μ).
Note: To obtain the specific values for the equation above, you would need to know the length of the wire, the length of the copper object below the water line, the tension in the wire, and the linear mass density of the wire.