Tuning a Cello. A cellist tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.625m long and has a mass of 15.0 G.
Tension: 160 N
What percent increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from C to D?
6.1
To find the percent increase in tension needed to increase the frequency from 65.4 Hz to 73.4 Hz, you can use the formula for the frequency of a vibrating string:
f = 1 / (2L) * sqrt(T / μ)
where:
- f is the frequency
- L is the length of the vibrating portion of the string
- T is the tension in the string
- μ is the linear mass density of the string (mass per unit length)
First, let's calculate the initial linear mass density (μ) of the string:
μ = mass / length
Given that the mass is 15.0 g (0.015 kg) and the length is 0.625 m:
μ = 0.015 kg / 0.625 m = 0.024 kg/m
Next, let's calculate the tension needed to achieve a frequency of 73.4 Hz:
T = (f * (2L))^2 * μ
Substituting the values:
T = (73.4 Hz * (2 * 0.625 m))^2 * 0.024 kg/m
T = (73.4 Hz * 1.25 m)^2 * 0.024 kg/m
T = (91.75)^2 * 0.024 kg/m
T = 8405.5625 * 0.024 kg/m
T = 201.732 kg m/s^2
Now, let's find the percent increase in tension:
Percent increase = ((New Tension - Old Tension) / Old Tension) * 100%
Old Tension = 160 N (given)
New Tension = 201.732 kg m/s^2 * 9.8 m/s^2 = 1978.2664 N
Percent increase = ((1978.2664 N - 160 N ) / 160 N) * 100%
Percent increase = 1111.4165%
Therefore, the tension needs to be increased by approximately 1111.42% to increase the frequency from 65.4 Hz to 73.4 Hz.
To calculate the percent increase in tension needed to increase the frequency from 65.4 Hz to 73.4 Hz, we need to use the relationship between the frequency and tension of a vibrating string.
The fundamental frequency of a vibrating string is given by the formula:
f = (1/2L) * √(T/μ)
where:
- f is the frequency of the string
- L is the length of the vibrating portion of the string
- T is the tension in the string
- μ is the linear mass density of the string
We can rearrange this formula to solve for the tension T:
T = (f² * μ * 4L²)
First, we need to calculate the tension T for the initial frequency of 65.4 Hz. We know the length of the string (0.625m), the frequency (65.4 Hz), and the mass (15.0 g). However, we need to convert the mass to linear mass density by dividing it by the length:
μ = (mass / length) = (15.0 g / 0.625 m)
Next, substitute the values into the formula to find the initial tension T:
T = (65.4² * (15.0 g / 0.625 m) * 4 * (0.625 m)²
Calculate the value of T for the initial frequency.
Next, we can calculate the tension T for the target frequency of 73.4 Hz using the same formula and the same values for length and linear mass density. Calculate the value of T for the target frequency.
Finally, to find the percent increase in tension, use the formula:
Percent increase = ((T_target - T_initial) / T_initial) * 100
Substitute the values of T_target and T_initial into the formula to calculate the percent increase in tension needed.