A wire 1m long and weighting 5 gm is stretched by a tension of 4kg wt. When sounded, it is found to vibrate in two loops. Calculate the frequency of the note emitted by the wire.
To calculate the frequency, we can use the formula for the frequency of a vibrating string:
Frequency (f) = (1/2L) * sqrt(T/μ)
Where:
L is the length of the wire
T is the tension in the wire
μ is the linear density of the wire
First, let's calculate the linear density (μ) of the wire using the given information. The linear density of a wire is defined as the mass per unit length.
Linear Density (μ) = mass (m) / length (L)
Given that the length (L) of the wire is 1m and the weight (mass) is 5gm, we need to convert the weight from grams to kilograms. Since 1kg = 1000g and 1gm = 0.001kg:
mass (m) = 5gm * 0.001kg/gm = 0.005kg
Now we can calculate the tension (T) using the given information. The tension is given in kilograms weight (kg wt), which needs to be converted to Newtons (N). The conversion factor is 1 kg wt = 9.8 N.
Tension (T) = 4kg wt * 9.8 N/kg wt = 39.2 N
Now we can substitute the values of tension (T) and linear density (μ) into the frequency formula:
Frequency (f) = (1/2L) * sqrt(T/μ)
Frequency (f) = (1/2 * 1) * sqrt(39.2 / 0.005)
Frequency (f) = 0.5 * sqrt(7840)
Now, we can solve for the frequency (f):
Frequency (f) = 0.5 * 88.49
Frequency (f) ≈ 44.25 Hz
Therefore, the frequency of the note emitted by the wire is approximately 44.25 Hz.