a wire under tension vibrates with a fundamental frequency of 256Hz. what would be the fundamental frequency if the wire were half as long, twice as thick, and under 1/4 tension?
f1=v1/L1=(1/L1)(T1/μ1)^1/2
f2=v2/L2=(1/L2)(T2/μ2)^1/2
then
f2/f1=(L1/L2)[T2μ1/(T1μ2)]^1/2
where
L2=1/2L1
T2=1/4T1
μ2=ρS2=4ρS1=4μ1
then
f2=1/2f1
f2=128 hz
The speed of transverse waves is the squareroot of (T/p), where p is the density per unit length, and T is the tension foorce. The density per unit length is 4x higher for the shorter, fatter wire, and T is 1/4 as high, to the sound speed is 1/4 as large.
The wavelength of the fundamental is 1/2 as long due to the shorter length.
The fundamental frequency scales with (sound speed)/(wavelength), and gets multiplied by (1/4)/(1/2) = 1/2
A WIRE UNDER TENSION VIBRATES WITH A FUNDAMENTAL FREQUENCY OF 256 HZ. WHAT WOULD BE
THE FUNDAMENTAL FREQUENCY IF THE WIRE WERE HALF AS LONG, TWICE AS THICK AND UNDER
ONE-FOURTH THE TENSION
Given: f1 = 256 Hz
L1 = ½ L1
T2 = ¼ T1
Find: f2
To find the fundamental frequency of the wire under the given conditions, we can use the equation for the fundamental frequency of a vibrating wire:
f = (1/2L) * √(T/μ),
where:
f is the frequency,
L is the length of the wire,
T is the tension in the wire, and
μ is the linear density of the wire.
In this case, we are given that the original wire has a fundamental frequency of 256 Hz. We need to determine the new fundamental frequency when the wire is half as long, twice as thick, and under 1/4 tension.
1. Length of the new wire (L):
Since the wire is now half as long, we can substitute L/2 for L in the equation.
2. Tension of the new wire (T):
The tension of the new wire is 1/4 of the original tension. Therefore, we can substitute T/4 for T in the equation.
3. Linear density of the new wire (μ):
Since the wire is now twice as thick, we can substitute 2μ for μ in the equation.
Applying these changes to the equation, we get:
f_new = (1/2(L/2)) * √((T/4)/(2μ))
Simplifying further:
f_new = (1/4L) * √(T/8μ)
Now, substitute the original frequency (f = 256 Hz) into the equation and solve for the new fundamental frequency (f_new):
256 Hz = (1/4L) * √(T/8μ)
To find f_new, we need to determine the values of L, T, and μ.